I'm trying to find the infinitesimal generator of this process
$dY_{t}=\dfrac{b-Y_{t}}{1-t}dt+dB_{t}$ $0\leq a <1$, $Y_{0}=a$
where $B_{t}$ is a Brownian motion; and I've found the solution:
$Y_{t}=a(t-1)+bt+(1-t)\int_{0}^{t}\dfrac{dB_{s}}{1-s}$
but I don't know how to find the infinitesimal generator because it's not homogeneous.
Can you help me please?
Thank you in advance
It is very similar to the usual one as explained here Lecture 10: Forward and Backward equations for SDEs
the generator is simply
$$\mathcal{L}f(x,t)=b(x,t)\cdot \nabla f(x,t)+\frac{1}{2}a(x,t)\nabla^2f(x,t),\text{ with } a(x,t):=\sigma(x,t)\sigma^{T}(x,t)$$
which follows by applying Itô's formula. So in the above case we have
$$\mathcal{L}f(x,t)=\frac{b-x}{1-t} f_{x}(x,t)+\frac{1}{2}f_{xx}(x,t).$$