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Xt is a weak solution to the SDE with dXt = ( −αXt + γ )dt + β dBt , ∀t ≥ 0 X0 = x0. α, β , and γ constants, and Bt is a brownina motion.

need to find the PDE for the transition density of X at time t, pt(x0,.), and solve it

the expectation is E[Xt]=γ /α +exp(-αt)(x0-γ/α) and the variance V[Xt]=b^2/(2α)(1-exp(-2αt).

and i know that Xt follows a normal distribution and that it has a stationary distribution. so p(t,x,y)=N~(E[Xt],V[Xt])

but how to find the PDE of the transition density/ probability distribution?

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It's an Ornstein-Uhlenbeck process. See en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process –  Byron Schmuland Apr 21 '11 at 23:47
    
@Byron Schmuland: i don't know how to find the pde of the probability distrybution –  lisa Apr 25 '11 at 1:36
    
The PDE is on the Wikipedia page that I linked to. It is the Fokker–Planck equation. –  Byron Schmuland Apr 25 '11 at 3:36

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