Let $X_t$ be a semi-martingale and $H_t$ be a predictable process and $g$ be a measurable bijective function with measurable inverse. Does there exist a function $f(h,x)$ satisfying $$ \int_0^Tf(H_t,X_t) dg(X_t) = \int_0^TH_t dX_t? $$

If not what conditions do we need for that to hold?