Second Order Ornstein-Uhlenbeck Process

topkatz · February 25th, 2008, 9:57 pm

Oops, it posted before I could write my request. Anyway, here goes:Suppose I have a fairly typical O-U process:, where W is standard Brownian Motion, and now I have a second process Y that depends on X in a similar fashion:. The solution for X is well-known:My question will be whether it is correct to "bootstrap" this approach to get the solution for Y. Unfortunately, it looks like I'll have to post my proposed solution for Y separately, because I'm getting error messages from the equation editor.-- TMK --

topkatz · February 25th, 2008, 10:21 pm

Okay, let's see if I can continue where I left off:.Then.Plugging in X and simplifying gives:.Assuming my algebra is correct, is this the right answer, or does the stochasticity of X trip me up somehow?Thanks!-- TMK --

Alan · February 26th, 2008, 2:34 pm

2nd line looks ok (haven't checked the third). The main point is that the Ito differential of I(t) = int_0^t f(s) X(s) ds behaves just like the ordinary differential: dI(t) = f(t) X(t) dt. That's because dt^2 = 0 in a stochastic calculus sense, even though X(t) may be a diffusion process.regards,

topkatz · February 26th, 2008, 3:42 pm

Hi Alan, thank you for responding!I think I can further reduce the double integral at the end of my final expression for Y as follows:Does that look right? Thanks!-- TMK --