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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. –  Did Nov 27 '12 at 22:09
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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. –  Stefan Hansen Nov 28 '12 at 14:38
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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. –  Did Nov 28 '12 at 19:54
    
Please refrain from useless unconstructive comments –  MathewG Nov 29 '12 at 19:51
    
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? –  Did Nov 29 '12 at 21:45

2 Answers 2

up vote 0 down vote accepted

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.

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