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Time-dependent Markov process: infinitesimal generator

If one talks about homogeneous Markov diffusion $$\mathrm dX_t = \mu(X_t)\mathrm dt+\sigma(X_t)\mathrm dw_t$$ with $\mu,\sigma$ sufficiently differentiable and of appropriate dimensions, there is nice equation for a function $m_f(x,t) = \mathsf E_x f(X_t)$ for $f\in C^2(\mathbb R)$: $$\begin{cases} \frac{\partial m_f}{\partial t} &= \mu\frac{\partial m_f}{\partial x}+\frac12\sigma^2\frac{\partial^2 m_f}{\partial x^2}, \ m(x,0)&=f(x). \end{cases}$$

On the other hand, while answering on the question Stochastic model I advised to use Fokker-Plank equation for the density of the process rather then the very same equation for $m_f$.

The problem I had is the following. Since $\mu$ and $\sigma$ are time-dependent there, one can construct a process $Z_t = (X_t,Y_t)$ with $\mathrm dY_t = \mathrm dt$ and obtain everything for this process just using theory of homogeneous/time-independent Markov processes (how it is usually written in the books). Unfortunately, if you derive an infinitesimal generator for this process then you obtain $$\mathcal A_Zg(x,y) = \frac{\partial g}{\partial y}+\mu\frac{\partial g}{\partial x}+\frac12\sigma^2\frac{\partial^2 g}{\partial x^2}.$$

For sure, now one shoud define $m_f(\tau|x,t) = \mathsf E_{x,t}f(X_{t+\tau})$ where $\tau\in \mathbb R_{\geq 0}$. If I am not wrong then it follows $$\frac{\partial m_f}{\partial \tau} = \frac{\partial m_f}{\partial t}+\mu\frac{\partial m_f}{\partial x}+\frac12\sigma^2\frac{\partial^2 m_f}{\partial x^2}\quad(*)$$ which is kind of strange equation.

My question has three parts:

1. is an equation $(*)$ correct? if yes, are there developed methods for its solutions?

2. if this equation is not correct, what is the right equation?

3. I usually have problems when deal with non-homogeneous Markov processes since trick $Z_t = (t,X_t)$ does not help me. Could you refer me to literature where authors consider non-homogeneous Markov processes in details (rather then saying that this trick will help to use provided theory of homogeneous processes)?

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You can directly deal with inhomogeneous Markov processes through the Kolmogorov backward and forward equations. I suppose you are asking for the forward equation (i.e. derivative with respect to the time in the future). Let me discuss things on a formal level, which means here I ignore regularity conditions to ensure the existence of generators, derivatives, etc etc.

Let me first give the backward equation, which is probably easier (a specific case of Feynman-Kac formula). Let $u(s,x,t)=\mathbb{E}^{s,x} f(X_t)$ for a nice function $f$ (e.g. infinitely differentiable with compact support), where the expectation is under the measure $\mathbb{P}^{s,x}$ such that $\mathbb{P}^{s,x}\{X_s=x\}=1$. Let $A_s$ be the generator (I am ignoring the detail of how $A_s$ is defined, you can probably figure it out yourself or see references I give below). Then the backward equation states that

$\frac{\partial u}{\partial s}+A_s u =0$.

The forward equation is usually formulated for the density $p(s,x,t,y)$ of the transition kernel $P_{s,t}(x,B)=\int_B p(s,x,t,y)dy$:

$\frac{\partial p}{\partial t}=A_t^* p(s,x,t,y)$

where $A_t^*$ is the adjoint of the operator $A_t$ (again, there's subtlety in how you definite $A_t$ for forward and backward equations separately, I ignore that here).

For a given diffusion given as an SDE $dX_t=\mu(t,X_t)dt+\sigma(t,X_t)dW_t$ for a Wiener process $W$, it is pretty straightforward that $A_t f(x)=\mu(t,x)f'(x)+\frac{1}{2}\sigma^2(t,x)f''(x)$.

All these are discussed to certain extent in "The Theory of Stochastic Processes, Vol II" by Gikhman and Skorokhod, and "Multidimensional Diffusion Processes" by Stroock and Varadhan (expressed in an integral form instead of derivatives).

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