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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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    Welcome to math.SE: since you are new, let me mention that, in order to get the best possible answers, it is helpful if you say what your thoughts on the problem are; this will prevent people from telling you things you already know, and help them give their answers at the right level. Please consider rewriting your post.2012-11-27
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    It seems like you're editing your post back and forth just to bump it, which you shouldn't. It will automatically be bumped by the site, if there is no activity.2012-11-28
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    Twelve edits so far and still not a single word about your thoughts on the question. Especially if you are preparing for an exam (as opposed to, say, having to answer this for a homework due tomorrow), this is strange.2012-11-28
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    Please conform to the ways this site is working. Does *useless* in your comment means that you do not care about the remarks made to you?2012-11-29
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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.