# What is the Feynman Kac Formula for the Merton model

I know that for the diffusion process $$X_t = \mu(t, X_t) dt + \sigma(t, X_t) dB_t,$$ the function $$u(t, x) = \mathbb{E}_{x, t}[e^{\int_t^T r(s, X_s) ds} g(X_T)]$$ with the boundary condition $$u(T, x) = \phi(x)$$ satisfies (from the Feynman Kac formula) $$u_t(t, x) + \mu(t, x)u_x(t, x) + \frac{\sigma^2(t, x)}{2}u_{xx}(t, x) - r(t, x)u(t, x) = 0$$ with $$u(T, x) = \phi(x)$$.

My question is what is the form of the Feynman Kac formula (PDE) if $$X_t = \mu(t, X_t) dt + \sigma(t, X_t) dB_t + (Y_t − 1)dN_t,$$ where $$N_t$$ is a Poisson process with parameter $$\lambda>0$$ and $$Y_t$$ is a sequence of i.i.d. random variables with log-normal distribution.

I'm just starting to learn about jump processes so I do not know how to obtain the PDE for above mentioned $$u(t, x)$$ function in case of the jump process $$X_t$$.