       Re: SDE's

• To: mathgroup at smc.vnet.net
• Subject: [mg25936] Re: SDE's
• From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
• Date: Fri, 10 Nov 2000 02:40:15 -0500 (EST)
• Organization: Universitaet Leipzig
• References: <8udmgo\$eiu@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Hi,

I suppose that your system mean

In[]:=deqn = { D[X[t], t] == a + b D[W[t], t],
D[X[t], t] == a(c - X[t]) + b D[W[t], t],
D[X[t], t]/X[t] == mu + sigma D[W[t], t],
D[X[t], t] == a(((sigma^2)/4a) - X[t]) + sigma Sqrt[X[t]] D[W[t],
t]}

(please, use the correct syntax in future questions because your
"notation"
has several ambiguities, that the Mathematica syntax has not)

In[]:=s1 = Solve[Equal @@ Subtract @@@ Transpose[List @@@ Take[deqn,
2]], X[t]]
Out[]={{X[t] -> -1 + c}}

form

In[]:=deqn1 = deqn /. s1[] /. X'[t] -> 0

Out[]={0 == a + b*Derivative[W][t],
0 == a + b*Derivative[W][t],
0 == mu + sigma*Derivative[W][t],
0 == a*(1 - c + (a*sigma^2)/4) + Sqrt[-1 + c]*sigma*
Derivative[W][t]}

it is to see that a==mu and sigma==b, otherwise there is no
solution.

With

In[]:= dsol = DSolve[deqn2[], W[t], t]

we get

Out[]={{W[t] -> -((mu*t)/sigma) + C}}

and

In[]:=eqn3 = deqn2 /. ( D[#, t] & /@ dsol[])

Out[]={True, True, True, 0 == -(Sqrt[-1 + c]*mu) +
mu*(1 - c + (mu*sigma^2)/4)}

and the two solutions for c

In[]:=Solve[eqn3[], c]
Out[]={{c -> (6 + mu*sigma^2 - 2*Sqrt[1 + mu*sigma^2])/4},
{c -> (6 + mu*sigma^2 + 2*Sqrt[1 + mu*sigma^2])/4}}

And you get the solutions

{X[t]->1-(6 + mu*sigma^2 - 2*Sqrt[1 + mu*sigma^2])/4,
W[t] -> -((mu*t)/sigma) + C,
a->mu,
b->sigma}

and

{X[t]->1-(6 + mu*sigma^2 + 2*Sqrt[1 + mu*sigma^2])/4,
W[t] -> -((mu*t)/sigma) + C,
a->mu,
b->sigma}

Regards
Jens

mot4201 at my-deja.com wrote:
>
> Hi all,
>
> I was wondering if anyone could share his (her) code for the exact
> solutions of the following SDE's
>
> 1. dXt=a dt+b dWt
> 2. dXt=a(c-Xt)dt+b dWt
> 3. dXt/Xt=mu dt+sigma dWt
> 4. dXt=a(((sigma^2)/4a)-Xt)dt+sigma sqrt(Xt) dWt
>
> If you could at least share the exact solutions , I would do the
> simulations my self. Thank you.
>
> Mark
>
> Sent via Deja.com http://www.deja.com/