Prove $v(t)$ is a Markov Process, find its' transition density and its' generator.

Question

If $v(t)$ is the velocity of the spherical particle is $v(t) = v(0)e^{-\alpha t} + \frac{\sigma}{m} \int_0^t{e^{-\alpha(t-s)}}dw(s)$ with mass $m$, radius $r$, $\alpha = \frac{\sigma \pi r \eta}{m}$ and $\sigma = \sqrt{2m\alpha KT}$. $w(s)$ is the Wiener process. The process $V(t)$ is supposed to be the Ornstein-Ulenbek process. But, I don't see how? The definition does not match the definition of the process in Wikipedia. And I am not sure how to go about proving that it is a Markov Process.

Answer 1

There is a mistake in your formula. The upper bound of integration in the stochastic integral has to be $t$ not $1$. Then, differantiating in $t$ you can easily check that you $v_{t}$ satisfies the OU-SDE. Apart, you do also find that formula on wikipedia.