Prove that the predictable It's formula is [begin{aligned}Fleft(t, X_{t} ight)= & Fleft(0, X_{0} ight)+int_{0}^{t} partial_{s} Fleft(s, X_{s}

Prove that the predictable Itô's formula is

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

where \(X_{s}=\left(X_{s}^{i} ; i=1, \ldots, d\right)\).