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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.


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