Ito formula for Jump-processes

Question

Hey here I found Ito formula for jump-processes (Proposition 20.13). It states that for Ito process of the form $$X_t+\int_0^tu_sdB_s+\int_0^tv_sds+\int_0^t\eta_sdY_s$$

where $Y_t$ is a compound Poisson process, we have $$f(X_t)=f(X_0)+\int_0^tv_sf'(X_s)ds+\int_0^tu_sf'(X_s)dB_s+\frac{1}{2}\int_0^t f''(X_s)|u_s|^2ds+\int_0^t(f(X_s)-f(X_{s-}))dN_s.$$

What we have to assume about processes $u_s, v_s, \eta_s$ and function $f$? Can we take function of two arguments $t$ and $x$ or it must be afucntion of $x$ only? Because I would like to make this statement precisely

Answer 1

You can deduce some of the properties directly from the statement about $f(X_t)$.

Firstly, $f$ has to be twice differentiable (otherwise $\frac{1}{2}\int_0^t f''(X_s)|u_s|^2ds$ would be undefined).

Secondly, $u_s$ has to be square-integrable (again, otherwise the integral above would be undefined).

Thirdly, $v_s$ has to be integrable (otherwise $\int_0^tv_sf'(X_s)ds$ would be undefined).

In addition, $u_t$ has to be measurable with respect to the filtration generated by $B_t$ (otherwise the integral $\int_0^tu_sdB_s$ would be undefined), and $\eta_t$ has to be measurable with respect to the filtration generated by $Y_t$ (otherwise the integral $\int_0^t\eta_sdY_s$ would be undefined).