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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.

(2) Does $X_{t}$ have a stationary distribution? and if so find it.

(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 ($\delta_{x^{*}}(x)=1$ if $x=x^{*}$, $\delta_{x^{*}}(x)=0$ if $x\neq x^{*}$), and $f_{2}(x)=x$.

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. :)

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    @MathewG: Many edits keep the question on the front page. A bounty is used to attract answers. These could be construed as impatience for an answer to a problem that is due soon. It is helpful to tell what you have tried and where you are having trouble. These are the issues that did is hinting at. His comments might be a bit obtuse, but they are not pointless.2012-11-30

2 Answers 2

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This link solves the first part of the question http://www.math.ku.dk/~susanne/StatDiff/Overheads1b.pdf

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(1) Look up the distinction between forward and backward PDEs for a diffusion.

(2) Consider $-a( X_t - \gamma/\alpha)$ for intuition. Solve explicitly and take $t\rightarrow \infty$

(3) Read up on the probabilistic interpretation of solutions to the diffusion.