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tw
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Joined: May 10th, 2002, 3:30 pm

stochastic calculus/Ito question

August 3rd, 2002, 7:22 pm

Hi,I'm puzzled by a feature of a model I've been working on. Wouldbe very grateful if any one had any pointers about a way to proceed.I'm not even sure if the description is well enough posed, buthere goes...Basically, a quantity S satisfies a stochastic process of the formds= f(s) dt + g(s) dWAnother quantity, satisfies the differential equation, dx=m(s,x) dtm is an elementary function.The ultimate objective to get the distribution of x at time t.Initially I thought that there would be a way to getthe x equation in the form of a stochastic process, butcan't find a simple way to do this.I suppose the stochastic process can be integrated to obtain something of the form, S=S(W_t), then substituted into the diff. eqn. for x, to give some kind of non-linear stochasticallyperturbed equation, then perhaps do a Taylor expansion,involving W_t and make some clever argument about discarding terms?Anyhow, thanks for any comments.Tom
 
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reza
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Joined: August 30th, 2001, 3:40 pm

stochastic calculus/Ito question

August 3rd, 2002, 7:30 pm

not sure if I understand wellif x=M(s,t)dx = [ dM/dt + dM/ds f(s) + 1/2 d2M/ds2 g2(s) ] dt + dM/ds g(s) dW so I guess we need to get fromdx=m(s,x) dtsomething likedx = h(s,t) dt + n(s,t) dWand identify the two equations above ...do you have the actual expression for dx=m(s,x) dt ?is there a way to get something like dx = h(s,t) dt + n(s,t) dW ???
 
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tw
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Joined: May 10th, 2002, 3:30 pm

stochastic calculus/Ito question

August 3rd, 2002, 9:13 pm

QuoteOriginally posted by: rezanot sure if I understand wellif x=M(s,t)dx = [ dM/dt + dM/ds f(s) + 1/2 d2M/ds2 g2(s) ] dt + dM/ds g(s) dW so I guess we need to get fromdx=m(s,x) dtsomething likedx = h(s,t) dt + n(s,t) dWand identify the two equations above ...do you have the actual expression for dx=m(s,x) dt ?is there a way to get something like dx = h(s,t) dt + n(s,t) dW ???Cheers for the response Reza.To fill in some of the blanks, the problem comes from a stochastic DP.There's an underlying price, S, that satisfies some kind of stochastic process.Also, there's the state of the system, x, (the details of what the system actuallyis aren't really relevant to this bit of the calculation). After calculating the DP, you get a policy function, that defines how you change the state with each combination of price and system state.Hence the equation, dx=m(s,x)dt, where m is the policy function.I suppose I could write the equations asds=f(s)dt+g(s)dW1dx=m(s,x)dt+dum.dW2where dum is a dummy volatility number then construct a 2D Kolmogorov equation in (s,x), solve this, then integrate over s to get the density in x. Then finally takethe limit dum->0. However I strongly suspect the limits won't exist. cheers,Tom
 
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reza
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Joined: August 30th, 2001, 3:40 pm

stochastic calculus/Ito question

August 3rd, 2002, 10:34 pm

sorry for being slow, but my question is can you solve the ODEdx=m(x,s) dtto getx=f(s,t) ?
 
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tw
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Posts: 592
Joined: May 10th, 2002, 3:30 pm

stochastic calculus/Ito question

August 4th, 2002, 9:46 am

QuoteOriginally posted by: rezasorry for being slow, but my question is can you solve the ODEdx=m(x,s) dtto getx=f(s,t) ?Sorry, it's me that's being slow.This is exactly the part of the problem that confuses me. I don't currently have an explicit form for m, but imagine that there are no complexities, in solving dx=m(x,s)dt for x=f(s,t), if you treat S as a parameter.However, by analogy with normal calculus, if the S equation were deterministic, I would solve for S(t), sub. that into the x equation to get dx=m(x,s(t))dt, and then this is a simple first order diff. eqn. with time-dependent coefficients, which can be solved. However, since the S equation is stochastic its solution is S=S(W_t) or something equivalent. It's at this point the stochastic nature of the problem confuses me. If I could write the changein x as relating to an increment dW, then I think it would be plain sailing. The fact that this doesn'tseem to be possible made me wonder if the whole thing was well posed.