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<title>Numerical Realizations of Brownian Motion and the sKdV</title>
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<div class=3DSection1>

<h1 align=3Dcenter style=3D'text-align:center'>Numerical Realizations of Br=
ownian <span
class=3DSpellE>Moton</span> and Gaussian White Noise for the Stochastic <sp=
an
class=3DSpellE>KdV</span> Equation</h1>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<h3><span class=3DGramE>The Wiener process or Brownian motion.</span></h3>

<p class=3DMsoNormal style=3D'text-align:justify;line-height:200%'><o:p>&nb=
sp;</o:p></p>

<p class=3DMsoNormal style=3D'line-height:200%'><b style=3D'mso-bidi-font-w=
eight:
normal'>Definition:</b><span style=3D'mso-spacerun:yes'>&nbsp; </span>A sta=
ndard
Wiener process <i style=3D'mso-bidi-font-style:normal'>W<sub>t</sub></i> <i
style=3D'mso-bidi-font-style:normal'>=3D {W<sub>t</sub>, t </i><i style=3D'=
mso-bidi-font-style:
normal'><span style=3D'font-family:Symbol;mso-ascii-font-family:"Times New =
Roman";
mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-fon=
t-family:
Symbol'><span style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>=
&sup3;</span></span>
0}</i>, on <i style=3D'mso-bidi-font-style:normal'>[0, T]</i> is a process =
with <i
style=3D'mso-bidi-font-style:normal'>W<sub>0</sub> =3D 0 </i>and with stati=
onary independent
increments such that for any <i style=3D'mso-bidi-font-style:normal'>0 </i>=
<i
style=3D'mso-bidi-font-style:normal'><span style=3D'font-family:Symbol;mso-=
ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman";mso-char-type:sym=
bol;
mso-symbol-font-family:Symbol'><span style=3D'mso-char-type:symbol;mso-symb=
ol-font-family:
Symbol'>&pound;</span></span> s &lt; t </i><i style=3D'mso-bidi-font-style:=
normal'><span
style=3D'font-family:Symbol;mso-ascii-font-family:"Times New Roman";mso-han=
si-font-family:
"Times New Roman";mso-char-type:symbol;mso-symbol-font-family:Symbol'><span
style=3D'mso-char-type:symbol;mso-symbol-font-family:Symbol'>&pound;</span>=
</span>
T</i>, the increment <i style=3D'mso-bidi-font-style:normal'>W<sub>t</sub>&=
#8211;W<sub>s</sub></i>
is a Gaussian random variable with mean zero and variance equal to <i
style=3D'mso-bidi-font-style:normal'>t-s</i>. That is<i style=3D'mso-bidi-f=
ont-style:
normal'>, <span class=3DGramE>E{</span>W<sub>t</sub>-W<sub>s</sub>} =3D 0</=
i>, and <span
class=3DSpellE><i style=3D'mso-bidi-font-style:normal'>Var</i></span><i
style=3D'mso-bidi-font-style:normal'>{W<sub>t</sub>-W<sub>s</sub>} =3D t-s<=
/i>.</p>

<p class=3DMsoNormal style=3D'line-height:200%'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'line-height:200%'>Formally, <span class=3DSpe=
llE><i
style=3D'mso-bidi-font-style:normal'>dW<sub>t</sub></i></span> is a Gaussian
random variable with mean zero and variance <span class=3DSpellE><span
class=3DGramE><i style=3D'mso-bidi-font-style:normal'>dt</i></span></span>.=
</p>

<h3>Realizations of the Wiener process</h3>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>One can use a Gaussian random generator to generate
realizations of Brownian motion:</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

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<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>First we show a simulation of Brownian motion.</p>

<p class=3DMsoNormal><span style=3D'mso-spacerun:yes'>&nbsp;</span><span
style=3D'font-size:9.0pt;font-family:"Courier New";mso-fareast-font-family:=
"Times New Roman"'><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'font-size:9.0pt;font-family:"Courier New";mso-fareast-font-family:=
"Times New Roman"'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><a
name=3D"INPUT_2"><span class=3DInput><span style=3D'mso-bidi-font-family:"T=
imes New Roman"'>%
Set Up Variables<o:p></o:p></span></span></a></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_2'><span class=3DSpellE><span class=3DGramE><sp=
an
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>dt</sp=
an></span></span></span></span><span
style=3D'mso-bookmark:INPUT_2'><span class=3DGramE><span class=3DInput><span
style=3D'mso-bidi-font-family:"Times New Roman"'>=3D</span></span></span></=
span><span
style=3D'mso-bookmark:INPUT_2'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'>1;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_2'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'>N=3D200;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_2'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'>t=3D<span class=3DGramE>zeros(</span>N,1);<o:p></o:p></s=
pan></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_2'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'><o:p>&nbsp;</o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_2'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'>% Generate Brownian <span class=3DGramE>Motion</span><o:=
p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_2'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'>r=3D<span class=3DSpellE><span class=3DGramE>randn</span=
></span><span
class=3DGramE>(</span>N,1);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_2'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'>w=3D<span class=3DGramE>zeros(</span>N,1);<o:p></o:p></s=
pan></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_2'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'><o:p>&nbsp;</o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_2'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'>w0=3D0;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_2'><span class=3DGramE><span class=3DInput><span
style=3D'mso-bidi-font-family:"Times New Roman"'>t(</span></span></span></s=
pan><span
style=3D'mso-bookmark:INPUT_2'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'>1)=3D1;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_2'><span class=3DGramE><span class=3DInput><span
style=3D'mso-bidi-font-family:"Times New Roman"'>w(</span></span></span></s=
pan><span
style=3D'mso-bookmark:INPUT_2'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'>1)=3Dw0;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_2'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'><o:p>&nbsp;</o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_2'><span class=3DGramE><span class=3DInput><span
style=3D'mso-bidi-font-family:"Times New Roman"'>for</span></span></span></=
span><span
style=3D'mso-bookmark:INPUT_2'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'> <span class=3DSpellE>i</span>=3D2:N<o:p></o:p></span></=
span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_2'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'><span style=3D'mso-spacerun:yes'>&nbsp; </span><span
class=3DGramE>t(</span><span class=3DSpellE>i</span>)=3D<span class=3DSpell=
E>i</span>;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_2'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'><span style=3D'mso-spacerun:yes'>&nbsp; </span><span
class=3DGramE>w(</span><span class=3DSpellE>i</span>)=3Dw(i-1)+<span class=
=3DSpellE>sqrt</span>(<span
class=3DSpellE>dt</span>)*r(<span class=3DSpellE>i</span>);<o:p></o:p></spa=
n></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_2'><span class=3DGramE><span class=3DInput><span
style=3D'mso-bidi-font-family:"Times New Roman"'>end</span></span></span></=
span><span
style=3D'mso-bookmark:INPUT_2'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'><span style=3D'mso-spacerun:yes'>&nbsp; </span><o:p></o:=
p></span></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_2'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'><o:p>&nbsp;</o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_2'><span class=3DGramE><span class=3DInput><span
style=3D'mso-bidi-font-family:"Times New Roman"'>plot(</span></span></span>=
</span><span
style=3D'mso-bookmark:INPUT_2'><span class=3DSpellE><span class=3DInput><sp=
an
style=3D'mso-bidi-font-family:"Times New Roman"'>t,w</span></span></span></=
span><span
style=3D'mso-bookmark:INPUT_2'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'>,'-');<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_2'><span class=3DGramE><span class=3DInput><span
style=3D'mso-bidi-font-family:"Times New Roman"'>title(</span></span></span=
></span><span
style=3D'mso-bookmark:INPUT_2'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'>'Simulation of a Wiener Process')<o:p></o:p></span></spa=
n></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_2'><span class=3DSpellE><span class=3DGramE><sp=
an
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>xlabel=
</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_2'><span class=3DGramE><span class=3DInput><span
style=3D'mso-bidi-font-family:"Times New Roman"'>(</span></span></span></sp=
an><span
style=3D'mso-bookmark:INPUT_2'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'>'time t')<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_2'><span class=3DSpe=
llE><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>ylabel</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_2'><span class=3DGramE><span class=3DInput><span
style=3D'mso-bidi-font-family:"Times New Roman"'>(</span></span></span></sp=
an><span
style=3D'mso-bookmark:INPUT_2'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'>'W(t)') </span></span></span><span
style=3D'mso-spacerun:yes'>&nbsp;</span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><a name=3D"OUTPUT_2"><span style=3D'font-size:9.0pt;fo=
nt-family:
"Courier New";mso-fareast-font-family:"Times New Roman"'><!--[if gte vml 1]=
><v:shape
 id=3D"_x0000_i1026" type=3D"#_x0000_t75" style=3D'width:4in;height:252pt'>
 <v:imagedata src=3D"stochastic_files/image003.emz" o:title=3D""/>
 <o:lock v:ext=3D"edit" aspectratio=3D"f"/>
</v:shape><![endif]--><![if !vml]><img width=3D384 height=3D336
src=3D"stochastic_files/image004.gif" v:shapes=3D"_x0000_i1026"><![endif]><=
span
style=3D'mso-spacerun:yes'>&nbsp;</span></span></a><span
style=3D'mso-spacerun:yes'>&nbsp;</span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'font-size:9.0pt;font-family:"Courier New";mso-fareast-font-family:=
"Times New Roman"'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'font-size:9.0pt;font-family:"Courier New";mso-fareast-font-family:=
"Times New Roman"'>We
can also look at the mean and variance of the Wiener process.<o:p></o:p></s=
pan></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'font-size:9.0pt;font-family:"Courier New";mso-fareast-font-family:=
"Times New Roman"'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'font-size:9.0pt;font-family:"Courier New";mso-fareast-font-family:=
"Times New Roman"'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal><a name=3D"INPUT_3"><span class=3DInput><span style=3D=
'mso-bidi-font-family:
"Times New Roman"'>N=3D200;<o:p></o:p></span></span></a></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_3'><span class=3DGramE><span class=3DInput><span
style=3D'mso-bidi-font-family:"Times New Roman"'>mean=3D</span></span></spa=
n></span><span
style=3D'mso-bookmark:INPUT_3'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'>zeros(N,1);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_3'><span class=3DSpellE><span class=3DGramE><sp=
an
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>var</s=
pan></span></span></span></span><span
style=3D'mso-bookmark:INPUT_3'><span class=3DGramE><span class=3DInput><span
style=3D'mso-bidi-font-family:"Times New Roman"'>=3D</span></span></span></=
span><span
style=3D'mso-bookmark:INPUT_3'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'>zeros(N,1);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_3'><span class=3DGra=
mE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>for</s=
pan></span></span></span><span
style=3D'mso-bookmark:INPUT_3'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'> j=3D1:N<span style=3D'mso-spacerun:yes'>&nbsp; </span><=
/span></span></span><span
style=3D'mso-bookmark:INPUT_3'><span class=3DInput><span style=3D'font-size=
:9.0pt;
mso-fareast-font-family:"Times New Roman";font-weight:normal'><o:p></o:p></=
span></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_3'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'><span style=3D'mso-spacerun:yes'>&nbsp; </span>r=3D<span
class=3DSpellE><span class=3DGramE>randn</span></span><span class=3DGramE>(=
</span>N,1);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_3'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'><span style=3D'mso-spacerun:yes'>&nbsp; </span>w=3D<span
class=3DGramE>zeros(</span>N,1);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_3'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'><span style=3D'mso-spacerun:yes'>&nbsp; </span><span
class=3DGramE>w(</span>1)=3Dw0;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_3'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'><span style=3D'mso-spacerun:yes'>&nbsp; </span><span
class=3DGramE>for</span> <span class=3DSpellE>i</span>=3D2:N<o:p></o:p></sp=
an></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_3'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'><span style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </sp=
an><span
class=3DGramE>w(</span><span class=3DSpellE>i</span>)=3Dw(i-1)+<span class=
=3DSpellE>sqrt</span>(<span
class=3DSpellE>dt</span>)*r(<span class=3DSpellE>i</span>);<o:p></o:p></spa=
n></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_3'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'><span style=3D'mso-spacerun:yes'>&nbsp; </span><span
class=3DGramE>end</span><span style=3D'mso-spacerun:yes'>&nbsp; </span><o:p=
></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_3'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'><span style=3D'mso-spacerun:yes'>&nbsp; </span><span
class=3DGramE>mean=3D</span><span class=3DSpellE>mean+w</span>;<o:p></o:p><=
/span></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_3'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'><span style=3D'mso-spacerun:yes'>&nbsp; </span><span
class=3DSpellE><span class=3DGramE>var</span></span><span class=3DGramE>=3D=
</span><span
class=3DSpellE>var+w</span>.*w;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_3'><span class=3DGramE><span class=3DInput><span
style=3D'mso-bidi-font-family:"Times New Roman"'>end</span></span></span></=
span><span
style=3D'mso-bookmark:INPUT_3'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'><o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_3'><span class=3DGramE><span class=3DInput><span
style=3D'mso-bidi-font-family:"Times New Roman"'>mean=3D</span></span></spa=
n></span><span
style=3D'mso-bookmark:INPUT_3'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'>mean/N;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_3'><span class=3DSpellE><span class=3DGramE><sp=
an
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>var</s=
pan></span></span></span></span><span
style=3D'mso-bookmark:INPUT_3'><span class=3DGramE><span class=3DInput><span
style=3D'mso-bidi-font-family:"Times New Roman"'>=3D</span></span></span></=
span><span
style=3D'mso-bookmark:INPUT_3'><span class=3DSpellE><span class=3DInput><sp=
an
style=3D'mso-bidi-font-family:"Times New Roman"'>var</span></span></span></=
span><span
style=3D'mso-bookmark:INPUT_3'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'>/N;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_3'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'><o:p>&nbsp;</o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_3'><span class=3DGramE><span class=3DInput><span
style=3D'mso-bidi-font-family:"Times New Roman"'>plot(</span></span></span>=
</span><span
style=3D'mso-bookmark:INPUT_3'><span class=3DSpellE><span class=3DInput><sp=
an
style=3D'mso-bidi-font-family:"Times New Roman"'>t,mean,'.',t,var</span></s=
pan></span></span><span
style=3D'mso-bookmark:INPUT_3'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'>,'- ');<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_3'><span class=3DGramE><span class=3DInput><span
style=3D'mso-bidi-font-family:"Times New Roman"'>title(</span></span></span=
></span><span
style=3D'mso-bookmark:INPUT_3'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'>['Mean and Variance of ' num2str(N) ' samples of standard
Wiener Process'])<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_3'><span class=3DSpellE><span class=3DGramE><sp=
an
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>xlabel=
</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_3'><span class=3DGramE><span class=3DInput><span
style=3D'mso-bidi-font-family:"Times New Roman"'>(</span></span></span></sp=
an><span
style=3D'mso-bookmark:INPUT_3'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'>'time t')<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'mso-layout-grid-align:none;text-autospace:non=
e'><span
style=3D'mso-bookmark:INPUT_3'><span class=3DSpellE><span class=3DGramE><sp=
an
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>ylabel=
</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_3'><span class=3DGramE><span class=3DInput><span
style=3D'mso-bidi-font-family:"Times New Roman"'>(</span></span></span></sp=
an><span
style=3D'mso-bookmark:INPUT_3'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'>'Mean and Variance')<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_3'><span class=3DGra=
mE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>legend=
(</span></span></span></span><span
style=3D'mso-bookmark:INPUT_3'><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'>'Mean','Variance',0) </span></span></span><span
style=3D'mso-spacerun:yes'>&nbsp;</span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><a name=3D"OUTPUT_3"><!--[if gte vml 1]><v:shape id=3D=
"_x0000_i1027"
 type=3D"#_x0000_t75" style=3D'width:4in;height:252pt'>
 <v:imagedata src=3D"stochastic_files/image005.emz" o:title=3D""/>
 <o:lock v:ext=3D"edit" aspectratio=3D"f"/>
</v:shape><![endif]--><![if !vml]><img width=3D384 height=3D336
src=3D"stochastic_files/image006.gif" v:shapes=3D"_x0000_i1027"><![endif]><=
span
style=3D'mso-spacerun:yes'>&nbsp;</span></a><span
style=3D'mso-spacerun:yes'>&nbsp;</span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>The standard Wiener process should have mean zero and
variance <i style=3D'mso-bidi-font-style:normal'>t&#8211;s </i>and this is
demonstrated by the figure above for <i style=3D'mso-bidi-font-style:normal=
'>s =3D
0</i>. For extremely large samples, the variance would be a straight line. =
</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<h3>Confirmation of <span class=3DSpellE>Wadati</span> Identities</h3>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>We want to confirm <sub><!--[if gte vml 1]><v:shape id=
=3D"_x0000_i1028"
 type=3D"#_x0000_t75" style=3D'width:123pt;height:15pt' o:ole=3D"">
 <v:imagedata src=3D"stochastic_files/image007.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D164 height=3D20
src=3D"stochastic_files/image008.gif" v:shapes=3D"_x0000_i1028"><![endif]><=
/sub><!--[if gte mso 9]><xml>
 <o:OLEObject Type=3D"Embed" ProgID=3D"Equation.DSMT4" ShapeID=3D"_x0000_i1=
028"
  DrawAspect=3D"Content" ObjectID=3D"_1184127393">
 </o:OLEObject>
</xml><![endif]--><span style=3D'mso-spacerun:yes'>&nbsp;</span>in <span
class=3DSpellE>Wadati's</span> paper. First, we check the case that <sub><!=
--[if gte vml 1]><v:shape
 id=3D"_x0000_i1029" type=3D"#_x0000_t75" style=3D'width:30.75pt;height:12p=
t' o:ole=3D"">
 <v:imagedata src=3D"stochastic_files/image009.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D41 height=3D16
src=3D"stochastic_files/image010.gif" v:shapes=3D"_x0000_i1029"><![endif]><=
/sub><!--[if gte mso 9]><xml>
 <o:OLEObject Type=3D"Embed" ProgID=3D"Equation.DSMT4" ShapeID=3D"_x0000_i1=
029"
  DrawAspect=3D"Content" ObjectID=3D"_1184127394">
 </o:OLEObject>
</xml><![endif]--><span style=3D'mso-spacerun:yes'>&nbsp;</span>Picking tim=
es <i
style=3D'mso-bidi-font-style:normal'>t =3D 5, 8</i>, we should get an answe=
r of 5. </p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><a name=3D"INPUT_17"><span class=3DInput><span style=
=3D'mso-bidi-font-family:
"Times New Roman"'>N=3D1000;<o:p></o:p></span></span></a></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_17'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>s=3D0;<o:p></o:p></span></=
span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_17'><span class=3DGr=
amE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>for</s=
pan></span></span></span><span
style=3D'mso-bookmark:INPUT_17'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'> j=3D1:N<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_17'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>r=3D<span class=3DSpel=
lE><span
class=3DGramE>randn</span></span><span class=3DGramE>(</span>N,1);<o:p></o:=
p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_17'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>w=3D<span class=3DGram=
E>zeros(</span>N,1);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_17'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span><span class=3DGramE>w(=
</span>1)=3Dw0;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_17'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span><span class=3DGramE>fo=
r</span>
<span class=3DSpellE>i</span>=3D2:N<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_17'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </spa=
n><span
class=3DGramE>w(</span><span class=3DSpellE>i</span>)=3Dw(i-1)+<span class=
=3DSpellE>sqrt</span>(<span
class=3DSpellE>dt</span>)*r(<span class=3DSpellE>i</span>);<o:p></o:p></spa=
n></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_17'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span><span class=3DGramE>en=
d</span><o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_17'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>% Note that t=3D5 corr=
esponds
to <span class=3DSpellE>i</span>=3D6<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_17'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>s=3D<span class=3DSpel=
lE>s+<span
class=3DGramE>w</span></span><span class=3DGramE>(</span>6)*w(9);<o:p></o:p=
></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_17'><span class=3DGr=
amE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>end</s=
pan></span></span></span><span
style=3D'mso-bookmark:INPUT_17'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'><span style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span><o:=
p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_17'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>s=3Ds/N </span></span></sp=
an><span
style=3D'mso-spacerun:yes'>&nbsp;</span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><a name=3D"OUTPUT_17"><span class=3DOutput><span
style=3D'mso-bidi-font-family:"Times New Roman"'>s =3D<o:p></o:p></span></s=
pan></a></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:OUTPUT_17'><span class=3DO=
utput><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>4.9122 </span></span><=
/span><span
style=3D'mso-spacerun:yes'>&nbsp;</span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>Next, we try a multiplier of <sub><!--[if gte vml 1]><=
v:shape
 id=3D"_x0000_i1030" type=3D"#_x0000_t75" style=3D'width:39pt;height:17.25p=
t' o:ole=3D"">
 <v:imagedata src=3D"stochastic_files/image011.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D52 height=3D23
src=3D"stochastic_files/image012.gif" v:shapes=3D"_x0000_i1030"><![endif]><=
/sub><!--[if gte mso 9]><xml>
 <o:OLEObject Type=3D"Embed" ProgID=3D"Equation.DSMT4" ShapeID=3D"_x0000_i1=
030"
  DrawAspect=3D"Content" ObjectID=3D"_1184127395">
 </o:OLEObject>
</xml><![endif]--><span style=3D'mso-spacerun:yes'>&nbsp;</span>in order to=
 give
the right expectation value for products of Wiener processes.<span
style=3D'mso-spacerun:yes'>&nbsp; </span>Let <sub><!--[if gte vml 1]><v:sha=
pe
 id=3D"_x0000_i1031" type=3D"#_x0000_t75" style=3D'width:30.75pt;height:12p=
t' o:ole=3D"">
 <v:imagedata src=3D"stochastic_files/image013.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D41 height=3D16
src=3D"stochastic_files/image014.gif" v:shapes=3D"_x0000_i1031"><![endif]><=
/sub><!--[if gte mso 9]><xml>
 <o:OLEObject Type=3D"Embed" ProgID=3D"Equation.DSMT4" ShapeID=3D"_x0000_i1=
031"
  DrawAspect=3D"Content" ObjectID=3D"_1184127396">
 </o:OLEObject>
</xml><![endif]--><span style=3D'mso-spacerun:yes'>&nbsp;</span>Then for <s=
ub><!--[if gte vml 1]><v:shape
 id=3D"_x0000_i1032" type=3D"#_x0000_t75" style=3D'width:24.75pt;height:15p=
t' o:ole=3D"">
 <v:imagedata src=3D"stochastic_files/image015.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D33 height=3D20
src=3D"stochastic_files/image016.gif" v:shapes=3D"_x0000_i1032"><![endif]><=
/sub><!--[if gte mso 9]><xml>
 <o:OLEObject Type=3D"Embed" ProgID=3D"Equation.DSMT4" ShapeID=3D"_x0000_i1=
032"
  DrawAspect=3D"Content" ObjectID=3D"_1184127397">
 </o:OLEObject>
</xml><![endif]-->and <sub><!--[if gte vml 1]><v:shape id=3D"_x0000_i1033"
 type=3D"#_x0000_t75" style=3D'width:27.75pt;height:15pt' o:ole=3D"">
 <v:imagedata src=3D"stochastic_files/image017.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D37 height=3D20
src=3D"stochastic_files/image018.gif" v:shapes=3D"_x0000_i1033"><![endif]><=
/sub><!--[if gte mso 9]><xml>
 <o:OLEObject Type=3D"Embed" ProgID=3D"Equation.DSMT4" ShapeID=3D"_x0000_i1=
033"
  DrawAspect=3D"Content" ObjectID=3D"_1184127398">
 </o:OLEObject>
</xml><![endif]--><span style=3D'mso-spacerun:yes'>&nbsp;</span>we should h=
ave <sub><!--[if gte vml 1]><v:shape
 id=3D"_x0000_i1034" type=3D"#_x0000_t75" style=3D'width:123.75pt;height:15=
pt' o:ole=3D"">
 <v:imagedata src=3D"stochastic_files/image019.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D165 height=3D20
src=3D"stochastic_files/image020.gif" v:shapes=3D"_x0000_i1034"><![endif]><=
/sub><!--[if gte mso 9]><xml>
 <o:OLEObject Type=3D"Embed" ProgID=3D"Equation.DSMT4" ShapeID=3D"_x0000_i1=
034"
  DrawAspect=3D"Content" ObjectID=3D"_1184127399">
 </o:OLEObject>
</xml><![endif]--><span style=3D'mso-spacerun:yes'>&nbsp;</span><span
class=3DGramE>Let's</span> see.</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><a name=3D"INPUT_19"><span class=3DInput><span style=
=3D'mso-bidi-font-family:
"Times New Roman"'>N=3D2000;<o:p></o:p></span></span></a></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_19'><span class=3DSp=
ellE><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>eps</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_19'><span class=3DGramE><span class=3DInput><sp=
an
style=3D'mso-bidi-font-family:"Times New Roman"'>=3D</span></span></span></=
span><span
style=3D'mso-bookmark:INPUT_19'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>0.1;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_19'><span class=3DSp=
ellE><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>mu</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_19'><span class=3DGramE><span class=3DInput><sp=
an
style=3D'mso-bidi-font-family:"Times New Roman"'>=3D</span></span></span></=
span><span
style=3D'mso-bookmark:INPUT_19'><span class=3DSpellE><span class=3DInput><s=
pan
style=3D'mso-bidi-font-family:"Times New Roman"'>sqrt</span></span></span><=
/span><span
style=3D'mso-bookmark:INPUT_19'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>(2*<span class=3DSpellE>eps</span>);<o:p></o:p></span></=
span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_19'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>s=3D0;<o:p></o:p></span></=
span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_19'><span class=3DGr=
amE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>for</s=
pan></span></span></span><span
style=3D'mso-bookmark:INPUT_19'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'> j=3D1:N<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_19'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>r=3D<span class=3DSpel=
lE><span
class=3DGramE>randn</span></span><span class=3DGramE>(</span>N,1);<o:p></o:=
p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_19'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>w=3D<span class=3DGram=
E>zeros(</span>N,1);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_19'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span><span class=3DGramE>w(=
</span>1)=3Dw0;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_19'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span><span class=3DGramE>fo=
r</span>
<span class=3DSpellE>i</span>=3D2:N<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_19'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </spa=
n><span
class=3DGramE>w(</span><span class=3DSpellE>i</span>)=3Dw(i-1)+<span class=
=3DSpellE>mu</span>*<span
class=3DSpellE>sqrt</span>(<span class=3DSpellE>dt</span>)*r(<span class=3D=
SpellE>i</span>);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_19'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span><span class=3DGramE>en=
d</span><o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_19'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>% Note that t=3D5 corr=
esponds
to <span class=3DSpellE>i</span>=3D6<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_19'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>s=3D<span class=3DSpel=
lE>s+<span
class=3DGramE>w</span></span><span class=3DGramE>(</span>6)*w(9);<o:p></o:p=
></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_19'><span class=3DGr=
amE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>end</s=
pan></span></span></span><span
style=3D'mso-bookmark:INPUT_19'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'><span style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span><o:=
p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_19'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>s=3Ds/N </span></span></sp=
an><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;</span><o:p></o:p></span></span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><a name=3D"OUTPUT_19"><span class=3DOutput><span
style=3D'mso-bidi-font-family:"Times New Roman"'>s =3D<o:p></o:p></span></s=
pan></a></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:OUTPUT_19'><span class=3DO=
utput><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>1.0296</span></span> <=
/span><span
style=3D'mso-spacerun:yes'>&nbsp;</span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>There is some deviation. We can check the mean over se=
veral
runs.</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><a name=3D"INPUT_21"><span class=3DInput><span style=
=3D'mso-bidi-font-family:
"Times New Roman"'>N=3D1000;<o:p></o:p></span></span></a></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_21'><span class=3DSp=
ellE><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>eps</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_21'><span class=3DGramE><span class=3DInput><sp=
an
style=3D'mso-bidi-font-family:"Times New Roman"'>=3D</span></span></span></=
span><span
style=3D'mso-bookmark:INPUT_21'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>0.1;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_21'><span class=3DSp=
ellE><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>mu</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_21'><span class=3DGramE><span class=3DInput><sp=
an
style=3D'mso-bidi-font-family:"Times New Roman"'>=3D</span></span></span></=
span><span
style=3D'mso-bookmark:INPUT_21'><span class=3DSpellE><span class=3DInput><s=
pan
style=3D'mso-bidi-font-family:"Times New Roman"'>sqrt</span></span></span><=
/span><span
style=3D'mso-bookmark:INPUT_21'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>(2*<span class=3DSpellE>eps</span>);<o:p></o:p></span></=
span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_21'><span class=3DGr=
amE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>mean=
=3D</span></span></span></span><span
style=3D'mso-bookmark:INPUT_21'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>0;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_21'><span class=3DGr=
amE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>for</s=
pan></span></span></span><span
style=3D'mso-bookmark:INPUT_21'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'> k=3D1:20<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_21'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span><span class=3DGramE>s(=
</span>k)=3D0;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_21'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span><span class=3DGramE>fo=
r</span>
j=3D1:N<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_21'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </spa=
n>r=3D<span
class=3DSpellE><span class=3DGramE>randn</span></span><span class=3DGramE>(=
</span>N,1);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_21'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </spa=
n>w=3D<span
class=3DGramE>zeros(</span>N,1);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_21'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </spa=
n><span
class=3DGramE>w(</span>1)=3Dw0;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_21'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp; </span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;</span><span class=3DGramE>for=
</span> <span
class=3DSpellE>i</span>=3D2:N<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_21'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;
</span><span class=3DGramE>w(</span><span class=3DSpellE>i</span>)=3Dw(i-1)=
+<span
class=3DSpellE>mu</span>*<span class=3DSpellE>sqrt</span>(<span class=3DSpe=
llE>dt</span>)*r(<span
class=3DSpellE>i</span>);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_21'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </spa=
n><span
class=3DGramE>end</span><o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_21'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </spa=
n>%
Note that t=3D5 corresponds to <span class=3DSpellE>i</span>=3D6<o:p></o:p>=
</span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_21'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </spa=
n><span
class=3DGramE>s(</span>k)=3Ds(k)+w(6)*w(9);<o:p></o:p></span></span></span>=
</p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_21'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span><span class=3DGramE>en=
d</span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span><o:p></o:p></span></span></s=
pan></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_21'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span><span class=3DGramE>s(=
</span>k)=3Ds(k)/N;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_21'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span><span class=3DGramE>me=
an=3D</span><span
class=3DSpellE>mean+s</span>(k);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_21'><span class=3DGr=
amE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>end</s=
pan></span></span></span><span
style=3D'mso-bookmark:INPUT_21'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'><o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_21'><span class=3DGr=
amE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>mean=
=3D</span></span></span></span><span
style=3D'mso-bookmark:INPUT_21'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>mean/20 </span></span></span><span class=3DInput><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;</span><o:p></o:p></span></span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><a name=3D"OUTPUT_21"></a><span class=3DGramE><span
style=3D'mso-bookmark:OUTPUT_21'><span class=3DOutput><span style=3D'mso-bi=
di-font-family:
"Times New Roman"'>mean</span></span></span></span><span style=3D'mso-bookm=
ark:
OUTPUT_21'><span class=3DOutput><span style=3D'mso-bidi-font-family:"Times =
New Roman"'>
=3D<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:OUTPUT_21'><span class=3DO=
utput><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>0.9971</span></span> <=
/span><span
style=3D'mso-spacerun:yes'>&nbsp;</span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>Now we check this expectation for general times with <=
span
class=3DSpellE><span class=3DGramE>dt</span></span> &lt; 1. Again, we will =
pick
times of 5 and 8 for a test pair.<span style=3D'mso-spacerun:yes'>&nbsp; </=
span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><a name=3D"INPUT_22"><span class=3DInput><span style=
=3D'mso-bidi-font-family:
"Times New Roman"'>N=3D2000;<o:p></o:p></span></span></a></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_22'><span class=3DSp=
ellE><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>eps</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_22'><span class=3DGramE><span class=3DInput><sp=
an
style=3D'mso-bidi-font-family:"Times New Roman"'>=3D</span></span></span></=
span><span
style=3D'mso-bookmark:INPUT_22'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>0.1;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_22'><span class=3DSp=
ellE><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>mu</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_22'><span class=3DGramE><span class=3DInput><sp=
an
style=3D'mso-bidi-font-family:"Times New Roman"'>=3D</span></span></span></=
span><span
style=3D'mso-bookmark:INPUT_22'><span class=3DSpellE><span class=3DInput><s=
pan
style=3D'mso-bidi-font-family:"Times New Roman"'>sqrt</span></span></span><=
/span><span
style=3D'mso-bookmark:INPUT_22'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>(2*<span class=3DSpellE>eps</span>);<o:p></o:p></span></=
span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_22'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>s=3D0;<o:p></o:p></span></=
span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_22'><span class=3DSp=
ellE><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>dt</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_22'><span class=3DGramE><span class=3DInput><sp=
an
style=3D'mso-bidi-font-family:"Times New Roman"'>=3D</span></span></span></=
span><span
style=3D'mso-bookmark:INPUT_22'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>0.1;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_22'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>t1=3D5;<o:p></o:p></span><=
/span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_22'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>t2=3D8;<o:p></o:p></span><=
/span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_22'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>k1=3D<span class=3DGramE>r=
ound(</span>t1/dt+1);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_22'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>k2=3D<span class=3DGramE>r=
ound(</span>t2/dt+1);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_22'><span class=3DGr=
amE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>for</s=
pan></span></span></span><span
style=3D'mso-bookmark:INPUT_22'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'> j=3D1:N<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_22'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>t<span class=3DGramE>=
=3D(</span>j-1)*<span
class=3DSpellE>dt</span>; <o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_22'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>r=3D<span class=3DSpel=
lE><span
class=3DGramE>randn</span></span><span class=3DGramE>(</span>N,1);<o:p></o:=
p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_22'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>w=3D<span class=3DGram=
E>zeros(</span>N,1);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_22'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span><span class=3DGramE>w(=
</span>1)=3Dw0;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_22'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span><span class=3DGramE>fo=
r</span>
<span class=3DSpellE>i</span>=3D2:N<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_22'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </spa=
n><span
class=3DGramE>w(</span><span class=3DSpellE>i</span>)=3Dw(i-1)+<span class=
=3DSpellE>mu</span>*<span
class=3DSpellE>sqrt</span>(<span class=3DSpellE>dt</span>)*r(<span class=3D=
SpellE>i</span>);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_22'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span><span class=3DGramE>en=
d</span><o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_22'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>s=3D<span class=3DSpel=
lE>s+<span
class=3DGramE>w</span></span><span class=3DGramE>(</span>k1)*w(k2);<o:p></o=
:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_22'><span class=3DGr=
amE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>end</s=
pan></span></span></span><span
style=3D'mso-bookmark:INPUT_22'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'><span style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span><o:=
p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_22'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>s=3Ds/N </span></span></sp=
an><span
style=3D'mso-spacerun:yes'>&nbsp;</span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><a name=3D"OUTPUT_22"><span class=3DOutput><span
style=3D'mso-bidi-font-family:"Times New Roman"'>s =3D<o:p></o:p></span></s=
pan></a></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:OUTPUT_22'><span class=3DO=
utput><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>1.0113</span></span> <=
/span><span
style=3D'mso-spacerun:yes'>&nbsp;</span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>The next thing to check is that <sub><!--[if gte vml 1=
]><v:shape
 id=3D"_x0000_i1035" type=3D"#_x0000_t75" style=3D'width:2in;height:27pt' o=
:ole=3D"">
 <v:imagedata src=3D"stochastic_files/image021.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D192 height=3D36
src=3D"stochastic_files/image022.gif" v:shapes=3D"_x0000_i1035"><![endif]><=
/sub><!--[if gte mso 9]><xml>
 <o:OLEObject Type=3D"Embed" ProgID=3D"Equation.DSMT4" ShapeID=3D"_x0000_i1=
035"
  DrawAspect=3D"Content" ObjectID=3D"_1184127400">
 </o:OLEObject>
</xml><![endif]--><span style=3D'mso-spacerun:yes'>&nbsp;</span></p>

<p class=3DMsoNormal><a name=3D"INPUT_23"><span class=3DInput><span style=
=3D'mso-bidi-font-family:
"Times New Roman"'>N=3D200;<o:p></o:p></span></span></a></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_23'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>M=3D1000;<o:p></o:p></span=
></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_23'><span class=3DSp=
ellE><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>eps</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_23'><span class=3DGramE><span class=3DInput><sp=
an
style=3D'mso-bidi-font-family:"Times New Roman"'>=3D</span></span></span></=
span><span
style=3D'mso-bookmark:INPUT_23'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>0.1;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_23'><span class=3DSp=
ellE><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>mu</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_23'><span class=3DGramE><span class=3DInput><sp=
an
style=3D'mso-bidi-font-family:"Times New Roman"'>=3D</span></span></span></=
span><span
style=3D'mso-bookmark:INPUT_23'><span class=3DSpellE><span class=3DInput><s=
pan
style=3D'mso-bidi-font-family:"Times New Roman"'>sqrt</span></span></span><=
/span><span
style=3D'mso-bookmark:INPUT_23'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>(2*<span class=3DSpellE>eps</span>);<o:p></o:p></span></=
span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_23'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>c=3D1.0;<o:p></o:p></span>=
</span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_23'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>s=3D0;<o:p></o:p></span></=
span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_23'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>w0=3D0;<o:p></o:p></span><=
/span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_23'><span class=3DSp=
ellE><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>dt</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_23'><span class=3DGramE><span class=3DInput><sp=
an
style=3D'mso-bidi-font-family:"Times New Roman"'>=3D</span></span></span></=
span><span
style=3D'mso-bookmark:INPUT_23'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>0.1;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_23'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>meanw2=3D<span class=3DGra=
mE>zeros(</span>N,1);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_23'><span class=3DSp=
ellE><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>meanew</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_23'><span class=3DGramE><span class=3DInput><sp=
an
style=3D'mso-bidi-font-family:"Times New Roman"'>=3D</span></span></span></=
span><span
style=3D'mso-bookmark:INPUT_23'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>zeros(N,1);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_23'><span class=3DGr=
amE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>for</s=
pan></span></span></span><span
style=3D'mso-bookmark:INPUT_23'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'> <span class=3DSpellE>i</span>=3D1:M<o:p></o:p></span></=
span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_23'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>w2=3D<span class=3DGra=
mE>zeros(</span>N,1);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_23'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span><span class=3DSpellE><=
span
class=3DGramE>ew</span></span><span class=3DGramE>=3D</span>zeros(N,1);<o:p=
></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_23'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span><span class=3DGramE>fo=
r</span>
j=3D1:N<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_23'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </spa=
n>t<span
class=3DGramE>=3D(</span>j-1)*<span class=3DSpellE>dt</span>; <o:p></o:p></=
span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_23'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </spa=
n>r=3D<span
class=3DSpellE><span class=3DGramE>randn</span></span><span class=3DGramE>(=
</span>N,1);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_23'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </spa=
n>w=3D<span
class=3DGramE>zeros(</span>N,1);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_23'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </spa=
n><span
class=3DGramE>w(</span>1)=3Dw0;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_23'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </spa=
n><span
class=3DGramE>for</span> <span class=3DSpellE>i</span>=3D2:N<o:p></o:p></sp=
an></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_23'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;
</span><span class=3DGramE>w(</span><span class=3DSpellE>i</span>)=3Dw(i-1)=
+<span
class=3DSpellE>mu</span>*<span class=3DSpellE>sqrt</span>(<span class=3DSpe=
llE>dt</span>)*r(<span
class=3DSpellE>i</span>);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_23'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </spa=
n><span
class=3DGramE>end</span><o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_23'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </spa=
n>w2=3Dw.*w;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_23'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </spa=
n><span
class=3DSpellE><span class=3DGramE>ew</span></span><span class=3DGramE>=3D<=
/span>exp(c*w);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_23'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span><span class=3DGramE>en=
d</span><o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_23'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>meanw2=3Dmeanw2+w2;<o:=
p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_23'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span><span class=3DSpellE><=
span
class=3DGramE>meanew</span></span><span class=3DGramE>=3D</span><span class=
=3DSpellE>meanew+ew</span>;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_23'><span class=3DGr=
amE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>end</s=
pan></span></span></span><span
style=3D'mso-bookmark:INPUT_23'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'><o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_23'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>meanw2=3Dmeanw2/M;<o:p></o=
:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_23'><span class=3DSp=
ellE><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>meanew</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_23'><span class=3DGramE><span class=3DInput><sp=
an
style=3D'mso-bidi-font-family:"Times New Roman"'>=3D</span></span></span></=
span><span
style=3D'mso-bookmark:INPUT_23'><span class=3DSpellE><span class=3DInput><s=
pan
style=3D'mso-bidi-font-family:"Times New Roman"'>meanew</span></span></span=
></span><span
style=3D'mso-bookmark:INPUT_23'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>/M;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_23'><span class=3DGr=
amE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>plot(<=
/span></span></span></span><span
style=3D'mso-bookmark:INPUT_23'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>meanw2,meanew,'- ',meanw2,exp(c^2/2*meanw2),'.')<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span></span></span></span><=
span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;</span><o:p></o:p></span></span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><a name=3D"OUTPUT_23"><span style=3D'font-family:"Cour=
ier New";
mso-bidi-font-family:"Times New Roman";color:red'><!--[if gte vml 1]><v:sha=
pe
 id=3D"_x0000_i1036" type=3D"#_x0000_t75" style=3D'width:4in;height:252pt'>
 <v:imagedata src=3D"stochastic_files/image023.emz" o:title=3D""/>
 <o:lock v:ext=3D"edit" aspectratio=3D"f"/>
</v:shape><![endif]--><![if !vml]><img width=3D384 height=3D336
src=3D"stochastic_files/image024.gif" v:shapes=3D"_x0000_i1036"><![endif]><=
span
class=3DError><span style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;</span></span></span></span></a><span
style=3D'mso-spacerun:yes'>&nbsp;</span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<h3>The Stochastic <span class=3DSpellE>KdV</span> <span class=3DSpellE>Sol=
iton</span></h3>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>We should next investigate the exact solution for the
stochastic <span class=3DSpellE>KdV</span> as presented in <span class=3DSp=
ellE>Wadati's</span>
paper. In that case we have <sub><!--[if gte vml 1]><v:shape id=3D"_x0000_i=
1037"
 type=3D"#_x0000_t75" style=3D'width:236.25pt;height:33pt' o:ole=3D"">
 <v:imagedata src=3D"stochastic_files/image025.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D315 height=3D44
src=3D"stochastic_files/image026.gif" v:shapes=3D"_x0000_i1037"><![endif]><=
/sub><!--[if gte mso 9]><xml>
 <o:OLEObject Type=3D"Embed" ProgID=3D"Equation.DSMT4" ShapeID=3D"_x0000_i1=
037"
  DrawAspect=3D"Content" ObjectID=3D"_1184127401">
 </o:OLEObject>
</xml><![endif]--><span style=3D'mso-spacerun:yes'>&nbsp;</span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>There are two possible things to check. First, one cou=
ld numerical
look at the exact solution for <sub><!--[if gte vml 1]><v:shape id=3D"_x000=
0_i1038"
 type=3D"#_x0000_t75" style=3D'width:27.75pt;height:15pt' o:ole=3D"">
 <v:imagedata src=3D"stochastic_files/image027.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D37 height=3D20
src=3D"stochastic_files/image028.gif" v:shapes=3D"_x0000_i1038"><![endif]><=
/sub><!--[if gte mso 9]><xml>
 <o:OLEObject Type=3D"Embed" ProgID=3D"Equation.DSMT4" ShapeID=3D"_x0000_i1=
038"
  DrawAspect=3D"Content" ObjectID=3D"_1184127402">
 </o:OLEObject>
</xml><![endif]--><span style=3D'mso-spacerun:yes'>&nbsp;</span>with the wh=
ite
noise simulated and then the solution averaged. Secondly, one can then eval=
uate
the infinite series and determine the amplitude dependence. This can be
compared to the stochastic <span class=3DSpellE>KdV</span> simulation of th=
e PDE
or the above average. </p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>First, we plot the exact <span class=3DSpellE>soliton<=
/span>
solution for the <span class=3DSpellE>KdV</span> equation.</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><a name=3D"INPUT_27"></a><span class=3DGramE><span
style=3D'mso-bookmark:INPUT_27'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>zeta=3D</span></span></span></span><span style=3D'mso-bo=
okmark:
INPUT_27'><span class=3DInput><span style=3D'mso-bidi-font-family:"Times Ne=
w Roman"'>1.0;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_27'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>N=3D200;<o:p></o:p></span>=
</span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_27'><span class=3DSp=
ellE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>Tsteps=
</span></span></span></span><span
style=3D'mso-bookmark:INPUT_27'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>=3D200;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_27'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>u=3D<span class=3DGramE>ze=
ros(</span>Tsteps,N+1);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_27'><span class=3DGr=
amE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>x0</sp=
an></span></span></span><span
style=3D'mso-bookmark:INPUT_27'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>=3D-5.0;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_27'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>a=3D-10;<o:p></o:p></span>=
</span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_27'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>b=3D50;<o:p></o:p></span><=
/span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_27'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>x=3D<span class=3DSpellE><=
span
class=3DGramE>linspace</span></span><span class=3DGramE>(</span>a,b,N+1);<o=
:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_27'><span class=3DSp=
ellE><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>dx</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_27'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>=3D(b-a)/N;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_27'><span class=3DSp=
ellE><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>dt</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_27'><span class=3DGramE><span class=3DInput><sp=
an
style=3D'mso-bidi-font-family:"Times New Roman"'>=3D</span></span></span></=
span><span
style=3D'mso-bookmark:INPUT_27'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>dx^3/4;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_27'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>t<span class=3DGramE>=3D(<=
/span>0:Tsteps-1)*<span
class=3DSpellE>dt</span>;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_27'><span class=3DGr=
amE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>for</s=
pan></span></span></span><span
style=3D'mso-bookmark:INPUT_27'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'> <span class=3DSpellE>i</span>=3D1:Tsteps;<o:p></o:p></s=
pan></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_27'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span><span class=3DGramE>fo=
r</span>
j=3D1:N+1;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_27'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </spa=
n><span
class=3DGramE>u(</span><span class=3DSpellE>i,j</span>)=3D2*zeta^2*(<span
class=3DSpellE>sech</span>(zeta*(x(j)-4*zeta^2*t(<span class=3DSpellE>i</sp=
an>)-x0))).^2;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_27'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span><span class=3DGramE>en=
d</span><o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_27'><span class=3DGr=
amE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>end</s=
pan></span></span></span><span
style=3D'mso-bookmark:INPUT_27'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'> <o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_27'><span class=3DGr=
amE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>mesh(<=
/span></span></span></span><span
style=3D'mso-bookmark:INPUT_27'><span class=3DSpellE><span class=3DInput><s=
pan
style=3D'mso-bidi-font-family:"Times New Roman"'>x,t,u</span></span></span>=
</span><span
style=3D'mso-bookmark:INPUT_27'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>)<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_27'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;</span></span></span></span><span
style=3D'mso-spacerun:yes'>&nbsp;</span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><a name=3D"OUTPUT_27"><!--[if gte vml 1]><v:shape id=
=3D"_x0000_i1039"
 type=3D"#_x0000_t75" style=3D'width:4in;height:252pt'>
 <v:imagedata src=3D"stochastic_files/image029.emz" o:title=3D""/>
 <o:lock v:ext=3D"edit" aspectratio=3D"f"/>
</v:shape><![endif]--><![if !vml]><img width=3D384 height=3D336
src=3D"stochastic_files/image030.gif" v:shapes=3D"_x0000_i1039"><![endif]><=
span
style=3D'mso-spacerun:yes'>&nbsp;</span></a><span
style=3D'mso-spacerun:yes'>&nbsp;</span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>Now, we can add the stochastic terms.</p>

<p class=3DMsoNormal><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </spa=
n></p>

<p class=3DMsoNormal><a name=3D"INPUT_29"></a><span class=3DGramE><span
style=3D'mso-bookmark:INPUT_29'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>zeta=3D</span></span></span></span><span style=3D'mso-bo=
okmark:
INPUT_29'><span class=3DInput><span style=3D'mso-bidi-font-family:"Times Ne=
w Roman"'>1.0;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>N=3D200;<o:p></o:p></span>=
</span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DSp=
ellE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>Tsteps=
</span></span></span></span><span
style=3D'mso-bookmark:INPUT_29'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>=3D200;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>u=3D<span class=3DGramE>ze=
ros(</span>Tsteps,N+1);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DGr=
amE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>x0</sp=
an></span></span></span><span
style=3D'mso-bookmark:INPUT_29'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>=3D-5.0;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>a=3D-10;<o:p></o:p></span>=
</span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>b=3D50;<o:p></o:p></span><=
/span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>x=3D<span class=3DSpellE><=
span
class=3DGramE>linspace</span></span><span class=3DGramE>(</span>a,b,N+1);<o=
:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DSp=
ellE><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>dx</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_29'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>=3D(b-a)/N;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DSp=
ellE><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>dt</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_29'><span class=3DGramE><span class=3DInput><sp=
an
style=3D'mso-bidi-font-family:"Times New Roman"'>=3D</span></span></span></=
span><span
style=3D'mso-bookmark:INPUT_29'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>dx^3/4;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>t<span class=3DGramE>=3D(<=
/span>0:Tsteps-1)*<span
class=3DSpellE>dt</span>;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><o:p>&nbsp;</o:p></span></=
span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DSp=
ellE><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>eps</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_29'><span class=3DGramE><span class=3DInput><sp=
an
style=3D'mso-bidi-font-family:"Times New Roman"'>=3D</span></span></span></=
span><span
style=3D'mso-bookmark:INPUT_29'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>0.1;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DSp=
ellE><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>mu</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_29'><span class=3DGramE><span class=3DInput><sp=
an
style=3D'mso-bidi-font-family:"Times New Roman"'>=3D</span></span></span></=
span><span
style=3D'mso-bookmark:INPUT_29'><span class=3DSpellE><span class=3DInput><s=
pan
style=3D'mso-bidi-font-family:"Times New Roman"'>sqrt</span></span></span><=
/span><span
style=3D'mso-bookmark:INPUT_29'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>(2*<span class=3DSpellE>eps</span>);<o:p></o:p></span></=
span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><o:p>&nbsp;</o:p></span></=
span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>r=3D<span class=3DSpellE><=
span
class=3DGramE>randn</span></span><span class=3DGramE>(</span>Tsteps,1);<o:p=
></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>w=3D<span class=3DGramE>ze=
ros(</span>Tsteps,1);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DSp=
ellE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>Iw</sp=
an></span></span></span><span
style=3D'mso-bookmark:INPUT_29'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>=3D<span class=3DGramE>zeros(</span>Tsteps,1);<o:p></o:p=
></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DGr=
amE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>w(</sp=
an></span></span></span><span
style=3D'mso-bookmark:INPUT_29'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>1)=3D0;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DSp=
ellE><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>Iw</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_29'><span class=3DGramE><span class=3DInput><sp=
an
style=3D'mso-bidi-font-family:"Times New Roman"'>(</span></span></span></sp=
an><span
style=3D'mso-bookmark:INPUT_29'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>1)=3Dw(1)/2*<span class=3DSpellE>dt</span>;<o:p></o:p></=
span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DGr=
amE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>for</s=
pan></span></span></span><span
style=3D'mso-bookmark:INPUT_29'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'> <span class=3DSpellE>i</span>=3D2:Tsteps<o:p></o:p></sp=
an></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span><span class=3DGramE>w(=
</span><span
class=3DSpellE>i</span>)=3Dw(i-1)+<span class=3DSpellE>mu</span>*<span clas=
s=3DSpellE>sqrt</span>(<span
class=3DSpellE>dt</span>)*r(<span class=3DSpellE>i</span>);<o:p></o:p></spa=
n></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span><span class=3DSpellE>I=
w</span>(<span
class=3DSpellE>i</span>)=3D<span class=3DSpellE>Iw</span>(i-1)+(w(i-1)+w(<s=
pan
class=3DSpellE>i</span>))/2*<span class=3DSpellE>dt</span>;<o:p></o:p></spa=
n></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DGr=
amE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>end</s=
pan></span></span></span><span
style=3D'mso-bookmark:INPUT_29'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'><o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><o:p>&nbsp;</o:p></span></=
span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DGr=
amE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>for</s=
pan></span></span></span><span
style=3D'mso-bookmark:INPUT_29'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'> <span class=3DSpellE>i</span>=3D1:Tsteps;<o:p></o:p></s=
pan></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span><span class=3DGramE>for</spa=
n>
j=3D1:N+1;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span>u(<span class=3DSpellE>i,j</=
span>)=3Dw(<span
class=3DSpellE>i</span>)+2*zeta^2*(<span class=3DSpellE>sech</span>(zeta*(x=
(j)-4*zeta^2*t(<span
class=3DSpellE>i</span>)-x0-6*<span class=3DSpellE>Iw</span>(<span class=3D=
SpellE>i</span>)))).^2;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span><span class=3DGramE>end</spa=
n><o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DGr=
amE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>end</s=
pan></span></span></span><span
style=3D'mso-bookmark:INPUT_29'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'> <o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DGr=
amE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>mesh(<=
/span></span></span></span><span
style=3D'mso-bookmark:INPUT_29'><span class=3DSpellE><span class=3DInput><s=
pan
style=3D'mso-bidi-font-family:"Times New Roman"'>x,t,u</span></span></span>=
</span><span
style=3D'mso-bookmark:INPUT_29'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>) </span></span><span style=3D'mso-spacerun:yes'>&nbsp;<=
/span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_29'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><o:p>&nbsp;</o:p></span></=
span></span></p>

<span style=3D'mso-bookmark:INPUT_29'></span>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><a name=3D"OUTPUT_29"><span style=3D'font-family:"Cour=
ier New";
mso-bidi-font-family:"Times New Roman";color:red'><!--[if gte vml 1]><v:sha=
pe
 id=3D"_x0000_i1040" type=3D"#_x0000_t75" style=3D'width:4in;height:252pt'>
 <v:imagedata src=3D"stochastic_files/image031.emz" o:title=3D""/>
 <o:lock v:ext=3D"edit" aspectratio=3D"f"/>
</v:shape><![endif]--><![if !vml]><img width=3D384 height=3D336
src=3D"stochastic_files/image032.gif" v:shapes=3D"_x0000_i1040"><![endif]><=
span
class=3DError><span style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;</span></span></span></span></a><span
style=3D'mso-spacerun:yes'>&nbsp;</span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>Next, we want to average over several realizations of =
the
solution of the stochastic <span class=3DSpellE>KdV</span>. </p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><a name=3D"INPUT_30"><span class=3DInput><span style=
=3D'mso-bidi-font-family:
"Times New Roman"'>M=3D200;<o:p></o:p></span></span></a></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_30'><span class=3DGr=
amE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>zeta=
=3D</span></span></span></span><span
style=3D'mso-bookmark:INPUT_30'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>1.0;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_30'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>N=3D200;<o:p></o:p></span>=
</span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_30'><span class=3DSp=
ellE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>Tsteps=
</span></span></span></span><span
style=3D'mso-bookmark:INPUT_30'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>=3D1000;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_30'><span class=3DGr=
amE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>x0</sp=
an></span></span></span><span
style=3D'mso-bookmark:INPUT_30'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>=3D-5.0;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_30'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>a=3D-10;<o:p></o:p></span>=
</span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_30'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>b=3D50;<o:p></o:p></span><=
/span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_30'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>x=3D<span class=3DSpellE><=
span
class=3DGramE>linspace</span></span><span class=3DGramE>(</span>a,b,N+1);<o=
:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_30'><span class=3DSp=
ellE><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>dx</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_30'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>=3D(b-a)/N;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_30'><span class=3DSp=
ellE><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>dt</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_30'><span class=3DGramE><span class=3DInput><sp=
an
style=3D'mso-bidi-font-family:"Times New Roman"'>=3D</span></span></span></=
span><span
style=3D'mso-bookmark:INPUT_30'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>dx^3/4;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_30'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>t<span class=3DGramE>=3D(<=
/span>0:Tsteps-1)*<span
class=3DSpellE>dt</span>;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_30'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><o:p>&nbsp;</o:p></span></=
span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_30'><span class=3DSp=
ellE><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>eps</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_30'><span class=3DGramE><span class=3DInput><sp=
an
style=3D'mso-bidi-font-family:"Times New Roman"'>=3D</span></span></span></=
span><span
style=3D'mso-bookmark:INPUT_30'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>0.1;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_30'><span class=3DSp=
ellE><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>mu</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_30'><span class=3DGramE><span class=3DInput><sp=
an
style=3D'mso-bidi-font-family:"Times New Roman"'>=3D</span></span></span></=
span><span
style=3D'mso-bookmark:INPUT_30'><span class=3DSpellE><span class=3DInput><s=
pan
style=3D'mso-bidi-font-family:"Times New Roman"'>sqrt</span></span></span><=
/span><span
style=3D'mso-bookmark:INPUT_30'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>(2*<span class=3DSpellE>eps</span>);<o:p></o:p></span></=
span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_30'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'><o:p>&nbsp;</o:p></span></=
span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_30'><span class=3DSp=
ellE><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>uu</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_30'><span class=3DGramE><span class=3DInput><sp=
an
style=3D'mso-bidi-font-family:"Times New Roman"'>=3D</span></span></span></=
span><span
style=3D'mso-bookmark:INPUT_30'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>zeros(Tsteps,N+1);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_30'><span class=3DGr=
amE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>for</s=
pan></span></span></span><span
style=3D'mso-bookmark:INPUT_30'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'> k=3D1:M<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'margin-left:.2in'><span style=3D'mso-bookmark=
:INPUT_30'><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>r=3D<s=
pan
class=3DSpellE><span class=3DGramE>randn</span></span><span class=3DGramE>(=
</span>Tsteps,1);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'margin-left:.2in'><span style=3D'mso-bookmark=
:INPUT_30'><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>w=3D<s=
pan
class=3DGramE>zeros(</span>Tsteps,1);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'margin-left:.2in'><span style=3D'mso-bookmark=
:INPUT_30'><span
class=3DSpellE><span class=3DInput><span style=3D'mso-bidi-font-family:"Tim=
es New Roman"'>Iw</span></span></span></span><span
style=3D'mso-bookmark:INPUT_30'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>=3D<span class=3DGramE>zeros(</span>Tsteps,1);<o:p></o:p=
></span></span></span></p>

<p class=3DMsoNormal style=3D'margin-left:.2in'><span style=3D'mso-bookmark=
:INPUT_30'><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>w(</span></span></span></span><span
style=3D'mso-bookmark:INPUT_30'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>1)=3D0;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'margin-left:.2in'><span style=3D'mso-bookmark=
:INPUT_30'><span
class=3DSpellE><span class=3DGramE><span class=3DInput><span style=3D'mso-b=
idi-font-family:
"Times New Roman"'>Iw</span></span></span></span></span><span style=3D'mso-=
bookmark:
INPUT_30'><span class=3DGramE><span class=3DInput><span style=3D'mso-bidi-f=
ont-family:
"Times New Roman"'>(</span></span></span></span><span style=3D'mso-bookmark=
:INPUT_30'><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>1)=3Dw=
(1)/2*<span
class=3DSpellE>dt</span>;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'margin-left:.2in'><span style=3D'mso-bookmark=
:INPUT_30'><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>for</span></span></span></span><span
style=3D'mso-bookmark:INPUT_30'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'> <span class=3DSpellE>i</span>=3D2:Tsteps<o:p></o:p></sp=
an></span></span></p>

<p class=3DMsoNormal style=3D'margin-left:.2in'><span style=3D'mso-bookmark=
:INPUT_30'><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span><span class=3DGramE>w(=
</span><span
class=3DSpellE>i</span>)=3Dw(i-1)+<span class=3DSpellE>mu</span>*<span clas=
s=3DSpellE>sqrt</span>(<span
class=3DSpellE>dt</span>)*r(<span class=3DSpellE>i</span>);<o:p></o:p></spa=
n></span></span></p>

<p class=3DMsoNormal style=3D'margin-left:.2in'><span style=3D'mso-bookmark=
:INPUT_30'><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span><span class=3DSpellE>I=
w</span>(<span
class=3DSpellE>i</span>)=3D<span class=3DSpellE>Iw</span>(i-1)+(w(i-1)+w(<s=
pan
class=3DSpellE>i</span>))/2*<span class=3DSpellE>dt</span>;<o:p></o:p></spa=
n></span></span></p>

<p class=3DMsoNormal style=3D'margin-left:.2in'><span style=3D'mso-bookmark=
:INPUT_30'><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>end</span></span></span></span><span
style=3D'mso-bookmark:INPUT_30'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'><o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'margin-left:.2in'><span style=3D'mso-bookmark=
:INPUT_30'><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'><o:p>&=
nbsp;</o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'margin-left:.2in'><span style=3D'mso-bookmark=
:INPUT_30'><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>u=3D<s=
pan
class=3DGramE>zeros(</span>Tsteps,N+1);<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'margin-left:.2in'><span style=3D'mso-bookmark=
:INPUT_30'><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>for</span></span></span></span><span
style=3D'mso-bookmark:INPUT_30'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'> <span class=3DSpellE>i</span>=3D1:Tsteps;<o:p></o:p></s=
pan></span></span></p>

<p class=3DMsoNormal style=3D'margin-left:.2in'><span style=3D'mso-bookmark=
:INPUT_30'><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span><span class=3DGramE>for</spa=
n>
j=3D1:N+1;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'margin-left:.2in'><span style=3D'mso-bookmark=
:INPUT_30'><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span>u(<span class=3DSpellE>i,j</=
span>)=3Dw(<span
class=3DSpellE>i</span>)+2*zeta^2*(<span class=3DSpellE>sech</span>(zeta*(x=
(j)-4*zeta^2*t(<span
class=3DSpellE>i</span>)-x0-6*<span class=3DSpellE>Iw</span>(<span class=3D=
SpellE>i</span>)))).^2;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'margin-left:.2in'><span style=3D'mso-bookmark=
:INPUT_30'><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span><span class=3DGramE>end</spa=
n><o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'margin-left:.2in'><span style=3D'mso-bookmark=
:INPUT_30'><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>end</span></span></span></span><span
style=3D'mso-bookmark:INPUT_30'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'> <o:p></o:p></span></span></span></p>

<p class=3DMsoNormal style=3D'margin-left:.2in'><span style=3D'mso-bookmark=
:INPUT_30'><span
class=3DSpellE><span class=3DGramE><span class=3DInput><span style=3D'mso-b=
idi-font-family:
"Times New Roman"'>uu</span></span></span></span></span><span style=3D'mso-=
bookmark:
INPUT_30'><span class=3DGramE><span class=3DInput><span style=3D'mso-bidi-f=
ont-family:
"Times New Roman"'>=3D</span></span></span></span><span style=3D'mso-bookma=
rk:INPUT_30'><span
class=3DSpellE><span class=3DInput><span style=3D'mso-bidi-font-family:"Tim=
es New Roman"'>uu+u</span></span></span></span><span
style=3D'mso-bookmark:INPUT_30'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>;<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_30'><span class=3DGr=
amE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>end</s=
pan></span></span></span><span
style=3D'mso-bookmark:INPUT_30'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'><o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_30'><span class=3DSp=
ellE><span
class=3DGramE><span class=3DInput><span style=3D'mso-bidi-font-family:"Time=
s New Roman"'>uu</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_30'><span class=3DGramE><span class=3DInput><sp=
an
style=3D'mso-bidi-font-family:"Times New Roman"'>=3D</span></span></span></=
span><span
style=3D'mso-bookmark:INPUT_30'><span class=3DSpellE><span class=3DInput><s=
pan
style=3D'mso-bidi-font-family:"Times New Roman"'>uu</span></span></span></s=
pan><span
style=3D'mso-bookmark:INPUT_30'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>/M;<o:p></o:p></span></span></span></p>

<span style=3D'mso-bookmark:INPUT_30'></span>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><a name=3D"INPUT_32"></a><span class=3DGramE><span
style=3D'mso-bookmark:INPUT_32'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>mesh(</span></span></span></span><span style=3D'mso-book=
mark:
INPUT_32'><span class=3DSpellE><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'>x,t,uu</span></span></span></span><span style=3D'mso-boo=
kmark:
INPUT_32'><span class=3DInput><span style=3D'mso-bidi-font-family:"Times Ne=
w Roman"'>)
</span></span></span><span style=3D'mso-spacerun:yes'>&nbsp;</span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><a name=3D"OUTPUT_32"><!--[if gte vml 1]><v:shape id=
=3D"_x0000_i1041"
 type=3D"#_x0000_t75" style=3D'width:4in;height:252pt'>
 <v:imagedata src=3D"stochastic_files/image033.emz" o:title=3D""/>
 <o:lock v:ext=3D"edit" aspectratio=3D"f"/>
</v:shape><![endif]--><![if !vml]><img width=3D384 height=3D336
src=3D"stochastic_files/image034.gif" v:shapes=3D"_x0000_i1041"><![endif]><=
span
style=3D'mso-spacerun:yes'>&nbsp;</span></a><span
style=3D'mso-spacerun:yes'>&nbsp;</span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>This is an average over 200 runs. We now look at the
amplitude as a function of the time.</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><a name=3D"INPUT_31"></a><span class=3DGramE><span
style=3D'mso-bookmark:INPUT_31'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>plot(</span></span></span></span><span style=3D'mso-book=
mark:
INPUT_31'><span class=3DSpellE><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'>t,max</span></span></span></span><span style=3D'mso-book=
mark:
INPUT_31'><span class=3DInput><span style=3D'mso-bidi-font-family:"Times Ne=
w Roman"'>(<span
class=3DSpellE>uu</span>'))</span></span></span><span class=3DInput><span
style=3D'mso-bidi-font-family:"Times New Roman"'><o:p></o:p></span></span><=
/p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><a name=3D"OUTPUT_31"><!--[if gte vml 1]><v:shape id=
=3D"_x0000_i1042"
 type=3D"#_x0000_t75" style=3D'width:4in;height:252pt'>
 <v:imagedata src=3D"stochastic_files/image035.emz" o:title=3D""/>
 <o:lock v:ext=3D"edit" aspectratio=3D"f"/>
</v:shape><![endif]--><![if !vml]><img width=3D384 height=3D336
src=3D"stochastic_files/image036.gif" v:shapes=3D"_x0000_i1042"><![endif]><=
span
style=3D'mso-spacerun:yes'>&nbsp;</span></a><span
style=3D'mso-spacerun:yes'>&nbsp;</span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><a name=3D"INPUT_34"><span class=3DInput><span style=
=3D'mso-bidi-font-family:
"Times New Roman"'>z=3D</span></span></a><span class=3DGramE><span
style=3D'mso-bookmark:INPUT_34'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>max(</span></span></span></span><span style=3D'mso-bookm=
ark:
INPUT_34'><span class=3DSpellE><span class=3DInput><span style=3D'mso-bidi-=
font-family:
"Times New Roman"'>uu</span></span></span></span><span style=3D'mso-bookmar=
k:
INPUT_34'><span class=3DInput><span style=3D'mso-bidi-font-family:"Times Ne=
w Roman"'>');<o:p></o:p></span></span></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_34'><span class=3DGr=
amE><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'>plot(<=
/span></span></span></span><span
style=3D'mso-bookmark:INPUT_34'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>log(t),log(z),'.','<span class=3DSpellE>MarkerSize</span=
>',
20) </span></span></span><span style=3D'mso-spacerun:yes'>&nbsp;</span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><a name=3D"OUTPUT_34"><span class=3DOutput><span
style=3D'mso-bidi-font-family:"Times New Roman"'>Warning: Log of zero.<o:p>=
</o:p></span></span></a></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:OUTPUT_34'><!--[if gte vml=
 1]><v:shape
 id=3D"_x0000_i1043" type=3D"#_x0000_t75" style=3D'width:4in;height:252pt'>
 <v:imagedata src=3D"stochastic_files/image037.emz" o:title=3D""/>
 <o:lock v:ext=3D"edit" aspectratio=3D"f"/>
</v:shape><![endif]--><![if !vml]><img width=3D384 height=3D336
src=3D"stochastic_files/image038.gif" v:shapes=3D"_x0000_i1043"><![endif]><=
span
style=3D'mso-spacerun:yes'>&nbsp;</span></span><span
style=3D'mso-spacerun:yes'>&nbsp;</span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>A linear log-log plot indicates a power law behavior. =
<span
class=3DSpellE>Restrictng</span> to a linear potion of the plot and doing a=
 fit
leads to the power law. Namely<span class=3DGramE>,<span
style=3D'mso-spacerun:yes'>&nbsp; </span>if</span> <sub><!--[if gte vml 1]>=
<v:shape
 id=3D"_x0000_i1044" type=3D"#_x0000_t75" style=3D'width:38.25pt;height:15.=
75pt'
 o:ole=3D"">
 <v:imagedata src=3D"stochastic_files/image039.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D51 height=3D21
src=3D"stochastic_files/image040.gif" v:shapes=3D"_x0000_i1044"><![endif]><=
/sub><!--[if gte mso 9]><xml>
 <o:OLEObject Type=3D"Embed" ProgID=3D"Equation.DSMT4" ShapeID=3D"_x0000_i1=
044"
  DrawAspect=3D"Content" ObjectID=3D"_1184127404">
 </o:OLEObject>
</xml><![endif]--><span style=3D'mso-spacerun:yes'>&nbsp;</span>then <sub><=
!--[if gte vml 1]><v:shape
 id=3D"_x0000_i1045" type=3D"#_x0000_t75" style=3D'width:75.75pt;height:15p=
t' o:ole=3D"">
 <v:imagedata src=3D"stochastic_files/image041.wmz" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D101 height=3D20
src=3D"stochastic_files/image042.gif" v:shapes=3D"_x0000_i1045"><![endif]><=
/sub><!--[if gte mso 9]><xml>
 <o:OLEObject Type=3D"Embed" ProgID=3D"Equation.DSMT4" ShapeID=3D"_x0000_i1=
045"
  DrawAspect=3D"Content" ObjectID=3D"_1184127405">
 </o:OLEObject>
</xml><![endif]--><span style=3D'mso-spacerun:yes'>&nbsp;</span>Thus, the s=
lope
give the power in the power law. </p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><a name=3D"INPUT_36"></a><span class=3DSpellE><span cl=
ass=3DGramE><span
style=3D'mso-bookmark:INPUT_36'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>loglog</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_36'><span class=3DGramE><span class=3DInput><sp=
an
style=3D'mso-bidi-font-family:"Times New Roman"'>(</span></span></span></sp=
an><span
style=3D'mso-bookmark:INPUT_36'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>t(100:400),z(100:400),'.','<span class=3DSpellE>MarkerSi=
ze</span>',
20)<span style=3D'mso-spacerun:yes'>&nbsp; </span></span></span></span><span
class=3DInput><span style=3D'mso-bidi-font-family:"Times New Roman"'><o:p><=
/o:p></span></span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><a name=3D"OUTPUT_36"><span style=3D'font-family:"Cour=
ier New";
mso-bidi-font-family:"Times New Roman";color:red'><!--[if gte vml 1]><v:sha=
pe
 id=3D"_x0000_i1046" type=3D"#_x0000_t75" style=3D'width:4in;height:252pt'>
 <v:imagedata src=3D"stochastic_files/image043.emz" o:title=3D""/>
 <o:lock v:ext=3D"edit" aspectratio=3D"f"/>
</v:shape><![endif]--><![if !vml]><img width=3D384 height=3D336
src=3D"stochastic_files/image044.gif" v:shapes=3D"_x0000_i1046"><![endif]><=
span
class=3DError><span style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;</span></span></span></span></a><span
style=3D'mso-spacerun:yes'>&nbsp;</span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>A linear fit can be done using the following. We find =
that
the power is -1.16.</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><a name=3D"INPUT_38"><span class=3DInput><span style=
=3D'mso-bidi-font-family:
"Times New Roman"'>p=3D</span></span></a><span class=3DSpellE><span class=
=3DGramE><span
style=3D'mso-bookmark:INPUT_38'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>polyfit</span></span></span></span></span><span
style=3D'mso-bookmark:INPUT_38'><span class=3DGramE><span class=3DInput><sp=
an
style=3D'mso-bidi-font-family:"Times New Roman"'>(</span></span></span></sp=
an><span
style=3D'mso-bookmark:INPUT_38'><span class=3DInput><span style=3D'mso-bidi=
-font-family:
"Times New Roman"'>log(t(100:400)),log(z(100:400)),1)<o:p></o:p></span></sp=
an></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:INPUT_38'><span class=3DIn=
put><span
style=3D'mso-bidi-font-family:"Times New Roman"'>plot(log(t(100:400)),log(z=
(100:400)),'.',log(t(100:400)),<span
class=3DSpellE>polyval</span>(<span class=3DSpellE>p,log</span>(t (100:400)=
)),'-','<span
class=3DSpellE>MarkerSize</span>', 20,'LineWidth',2) </span></span></span><=
span
style=3D'mso-spacerun:yes'>&nbsp;</span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><a name=3D"OUTPUT_38"><span class=3DOutput><span
style=3D'mso-bidi-font-family:"Times New Roman"'>p =3D<o:p></o:p></span></s=
pan></a></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:OUTPUT_38'><span class=3DO=
utput><span
style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span>-1.1645<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span>-0.0744<o:p></o:p></span></s=
pan></span></p>

<p class=3DMsoNormal><span style=3D'mso-bookmark:OUTPUT_38'><span style=3D'=
font-family:
"Courier New";mso-bidi-font-family:"Times New Roman";color:red'><!--[if gte=
 vml 1]><v:shape
 id=3D"_x0000_i1047" type=3D"#_x0000_t75" style=3D'width:4in;height:252pt'>
 <v:imagedata src=3D"stochastic_files/image045.emz" o:title=3D""/>
 <o:lock v:ext=3D"edit" aspectratio=3D"f"/>
</v:shape><![endif]--><![if !vml]><img width=3D384 height=3D336
src=3D"stochastic_files/image046.gif" v:shapes=3D"_x0000_i1047"><![endif]><=
span
class=3DError><span style=3D'mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;</span></span></span></span></span><span
style=3D'mso-spacerun:yes'>&nbsp;</span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

</div>

</body>

</html>

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