Wat is Jump-diffusion, semimartingale?

A jump-diffusion is $X_t = X_0 + \int_0^t\mu_s\,ds + \int_0^t\sigma_s\,dW_s + J_t$, with $J$ a compound Poisson process (finitely many jumps on bounded intervals). A semimartingale is a process that is a local martingale plus an adapted process of finite variation, both càdlàg: the widest class of integrators for which stochastic integrals and Itô’s formula work.