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Math Help - Stochastic Differencial Equations

  1. #1
    nek
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    Stochastic Differencial Equations

    Hello, I'm trying to complete a course on SDE and I need to solve two stochastic differential equations.

    They are supposed to be easy, but I'm still a beginner and to be honest I'm quite stuck .

    The equations are the following:

    dXt = dt + dWtx

    dYt = XtdWty

    Where dWtxand dWtyare uncorrelated Brownian motions.

    Any answers or suggestions will be greatly appreciated! Thanks a lot!!
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  2. #2
    MHF Contributor
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    Re: Stochastic Differencial Equations

    Hey nek.

    First you need to solve for X(t) and then substitute in later for the SDE of Y(t) (When I use () I mean the way you have done above).

    So when you integrate both sides you get X(t) - X(0) = Integral_term_with_respect_to_t and Integral_term_with_respect_to_Wt where both have limits from 0 to t.

    Ito's Lemma has a generalized form to turn things like dX = f(X,Wt)dt + g(X,Wt)dWt into the above integral (where you have the two terms above). Since your f and g functions are pretty simple, you will get something reasonably simple.

    I'll wait for your response, but for the first one you only have integral of 1 with respect to dt and integral of 1 with respect to dWt. In this case what is integral of 1*du from 0 to t and integral 1*dWt from 0 to t? (Hint If I sum up all the changes of dWt from W(0) to W(t), what do I get)?
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