Suppose all processes and functions defined are nice enough such that all the following definitions make sense.

On a probability space $(\Omega,\mathcal{F},\mathbb{P})$ equipped with a filtration $\mathbb{F}$ satisfying the usual conditions of right continuity and $\mathbb{P}$ completeness, let $M$ be a continuous $\mathbb{F}$ martingale and $A$ a continuous $\mathbb{F}$ adapted process of finite variation. Now let $(E,\mathcal{B}(E))$ be a measurable Lusin space. Now let $\mu$ be a random measure on the space $(\Omega\times E,\mathcal{F}\times \mathcal{B}(E))$ and $\nu$ its compensator. For $\mathcal{P}$ measurable processes $\phi$ and $\psi$, and a $\mathcal{P}\otimes\mathcal{B}(E)$ measurable process $\beta(\cdot)$. Define the semi-martingale $X$ by $$X_t = X_0 + \int_0^t\phi_s dA_s + \int_0^t\psi_sdM_s + \int_0^t\int_E \beta_s(e)(\mu(ds,de)-\nu(ds,de)).$$

Now suppose $f$ is a twice continuously differentiable function, how do I apply Ito's formula to get the semi-martingale decomposition of $f(X_t)$? i.e. how do I calculate $df(X_t) = ...?$.


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