Derivation of a differential equation from a SDE

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...