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[[1]] /. X'[t] -> 0
Out[]={0 == a + b*Derivative[1][W][t],
0 == a + b*Derivative[1][W][t],
0 == mu + sigma*Derivative[1][W][t],
0 == a*(1 - c + (a*sigma^2)/4) + Sqrt[-1 + c]*sigma*
Derivative[1][W][t]}
it is to see that a==mu and sigma==b, otherwise there is no
solution.
With
In[]:= dsol = DSolve[deqn2[[3]], W[t], t]
we get
Out[]={{W[t] -> -((mu*t)/sigma) + C[1]}}
and
In[]:=eqn3 = deqn2 /. ( D[#, t] & /@ dsol[[1]])
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[[4]], 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[1],
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[1],
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/
> Before you buy.