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Suppose there is a non-homogeneous Markov process with state space $\mathbb{R}_{+}$ driven by this McKean-Vlasov-tipe SDE:

$$ dY_t = a \mathbb{E}[Y_t]\ dt - b\ Y_t\ dt - Y_t\ dN_{aY_t}$$

where $N_c$ is a Poisson point process with rate $c\geqslant 0$, and $a,b>0$.

I would like to know the strategy for deriving a differential equation for $v(x,t):=\mathbb{P}(Y_t\leqslant x)$ of the form

$$ \frac{\partial\ v}{\partial\ dt}(x,t) =...$$

any hints will be more than welcome.

I've thought that if I can compute $\mathbb{E}[f(Y_t)]$ for a wide range of functions, then I'd be able to compute the time-derivative of $v(x,t)$ somehow...

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