how to find derivative of a stochastic process?

Question

Consider the following equation for $X(t)$:

$$X(t)=e^{-bt}X(0)+\sigma\int_{0}^{b}e^{-b(t-s)}dW(t) \, ,$$

where $0 < b, \sigma\in\mathbb{R} $, $X(0)$ is the initial distribution of $X(t)$, independent of the Brownian motion $W(t)$. I want so show that $dX(t)= -bX(t)dt+\sigma dW(t)$, but I am getting stuck on computing the derivative of $$\sigma\int_{0}^{b}e^{-b(t-s)}dW(t) \, .$$

Could someone please give me some ideas? Thanks so much for your tim.

PS. the above equation is one of type of the Langevin's equation, more detail could be found here http://en.wikipedia.org/wiki/Langevin_equation

Answer 1

If you interpret the stochastic integral in the Ito-sense (often used in finance) you'll have to use Ito's lemma to evaluate it:
See e.g. here: Ito's lemma

Alternatively you could interpret it in the Stratonovich-sense (often used in physics):
See e.g. here: Stratonovich integral

A good introduction to solving these kinds of stochastic differential equations (sde) without the use of measure theory and with lots of intuition is e.g. Wiersema: Brownian motion calculus

Answer 2

Ok here is the trick

Use Itô's lemma mentioned by vonjd to the function $f(t,X_t)=e^{bt}.X(t)$ and after some algebra you'll get what you want.

Regards