Let \(N\) be an inhomogeneous Poisson process with intensity \(\lambda(t), M\) its compensated martingale, \(W\) a Brownian motion and

\[d X_{t}=h_{t} d t+f_{t} d W_{t}+g_{t} d M_{t},\]

where \(f, g\) and \(h\) are (bounded) predictable processes. Using the identity \(\sum_{s \leq t} \phi_{s} \Delta N_{s}=\int_{0}^{t} \phi_{s} d N_{s}\), prove that

\[\begin{aligned}
F\left(t, X_{t}\right)= & F\left(0, X_{0}\right)+\int_{0}^{t} \partial_{s} F\left(s, X_{s}\right) d s+\int_{0}^{t} \partial_{x} F\left(s, X_{s}\right) h_{s} d s \\
& +\frac{1}{2} \int_{0}^{t} \partial_{x x} F\left(s, X_{s}\right) f_{s}^{2} d s \\
& +\int_{0}^{t}\left[F\left(s, X_{s}+g_{s}\right)-F\left(s, X_{s}\right)-\partial_{x} F\left(s, X_{s}\right) g_{s}\right] \lambda_{s} d s \\
& +\int_{0}^{t} \partial_{x} F\left(s, X_{s}\right) f_{s} d W_{s}+\int_{0}^{t}\left[F\left(s, X_{s^{-}}+g_{s}\right)-F\left(s, X_{s^{-}}\right)\right] d M_{s} .
\end{aligned}\]