how to find derivative of a stochastic process? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T17:04:13Z http://mathoverflow.net/feeds/question/26895 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/26895/how-to-find-derivative-of-a-stochastic-process how to find derivative of a stochastic process? Steven 2010-06-03T03:55:33Z 2010-06-03T15:06:40Z <p>it is really shame on me that I cannot find the derivative of the following integral :(</p> <p>$$X(t)=e^{-bt}X(0)+\sigma\int_{0}^{b}e^{-b(t-s)}dW(t)$$</p> <p>Find $dX(t)$ ? where $0 &lt; b, \sigma\in\mathbb{R}$, $X(0)$ is 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 $$\sigma\int_{0}^{b}e^{-b(t-s)}dW(t)$$</p> <p>could some one please give me some ideas ? thanks so much for your time </p> <p>PS. the above equation is one of type of the Langevin's equation, more detail could be found here <a href="http://en.wikipedia.org/wiki/Langevin_equation" rel="nofollow">http://en.wikipedia.org/wiki/Langevin_equation</a></p> http://mathoverflow.net/questions/26895/how-to-find-derivative-of-a-stochastic-process/26902#26902 Answer by vonjd for how to find derivative of a stochastic process? vonjd 2010-06-03T09:24:34Z 2010-06-03T09:24:34Z <p>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: <br>See e.g. here: <a href="http://en.wikipedia.org/wiki/Ito_lemma" rel="nofollow">Ito's lemma</a></p> <p>Alternatively you could interpret it in the Stratonovich-sense (often used in physics): <br>See e.g. here: <a href="http://en.wikipedia.org/wiki/Stratonovich_integral" rel="nofollow">Stratonovich integral</a></p> <p>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: <a href="http://books.google.com/books?id=kUFdAQAACAAJ&amp;dq=wiersema&amp;hl=de&amp;ei=_XMHTMGYEdaJ_gabpeEE&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=20&amp;ved=0CIEBEOgBMBM" rel="nofollow">Brownian motion calculus</a></p> http://mathoverflow.net/questions/26895/how-to-find-derivative-of-a-stochastic-process/26930#26930 Answer by The Bridge for how to find derivative of a stochastic process? The Bridge 2010-06-03T15:06:40Z 2010-06-03T15:06:40Z <p>Ok here is the trick </p> <p>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.</p> <p>Regards</p>