Let $B$ be a one-dimensional Brownian motion with $B_0=0$. Let $f$ be a twice continuously differentiable function on $R$ and let $g$ be a continuous function on $R$. If the process $$ X_t=X_0+\int_0^t (\frac 12 f''(B_s)-f(B_s)g(B_s))e^{-V_s}ds+\int_0^t f'(B_s)e^{-V_s}dB_s $$ where $V_t=\int_0^T g(B_s)ds$.
(1) How to show that $X_t$ is a semi-martingale?
(2) Show that $X$ is a continuous local martingale if and only if $f''=2gf$.
(1) We know that $\int_0^t f(s)dB_s$ is a local martingale, then the second part is a local martingale. But how about the first part? I try to show that it is finite variation.
(2) It is clear that if $f''=2gf$, then $X$ is a local martingale. But how about the converse statement?