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 (Speechless).
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!!
Re: Stochastic Differencial Equations
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)?