Transition density and distribution: (Ornsteinâ€“Uhlenbeck process) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T01:47:05Z http://mathoverflow.net/feeds/question/114946 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/114946/transition-density-and-distribution-ornsteinuhlenbeck-process Transition density and distribution: (Ornsteinâ€“Uhlenbeck process) MemT 2012-11-30T01:01:26Z 2012-11-30T01:01:26Z <p>Let $\left(X_{t},\, t\geq0\right)$ be the weak solution to the SDE below with $\alpha,\,\beta,\,\gamma$ constants: $$dX_{t}=(-\alpha X_{t}+\gamma)dt+\beta dB_{t}\quad\forall t\geq0,\, X_{0}=x_{0}$$ (1) Let $p_{t}(x_{0},\cdot)$ be the transition density for $X$ at time $t$. Find the partial differential equation (PDE) for $p_{t}\left(x_{0},\cdot\right)$ and solve.</p> <p>(2) Does $X_{t}$ have a stationary distribution? and if so find it.</p> <p>(3) Using stochastic methods find explicit solution to each of the two: $i=1,\,2$ initial value problems: $$\partial_{t}u(t,x)=\frac{1}{2}\beta^{2}\partial_{xx}^{2}u(t,x)+\left(-\alpha x+\gamma\right)\partial_{x}u(t,x),$$ and $u(0,x)=f_{i}(x)$ where $f_{1}(x)=\delta_{x^{*}}(x)$ is the Dirac function ($\delta_{x^{<em>}}(x)=1$ if $x=x^{</em>}$, $\delta_{x^{<em>}}(x)=0$ if $x\neq x^{</em>}$), and $f_{2}(x)=x$.</p> <p>I came accross the above problem while preparing for my SDE exam. It was on a past paper. I would be grateful to someone who can clearly explain to me the solution process. :)</p>