Jtdylktuy

Let

• 
  $(Omega,mathcal A,operatorname P)$ be a complete probability space


• 
  $(mathcal F_t)_{tge0}$ be a filtration on $(Omega,mathcal A)$
  


• 
  $W$ be an $mathcal F$-Brownian motion on $(Omega,mathcal A,operatorname P)$
  


• 
  $X$ be a continuous $mathcal F$-adapted process on $(Omega,mathcal A,operatorname P)$ with $$X_t=X_0+int_0^tmu_s:{rm d}s+int_0^tsigma_s:{rm d}W_s;;;text{for all }tge0text{ almost surely}tag1$$ for some $mathcal F$-progressive processes $mu,sigma$ with $$int_0^t|mu_s|+|sigma_s|^2:{rm d}s<infty;;;text{almost surely for all }tge0tag2$$
  

Now, let $fin C^2(mathbb R)$. By the Itō formula, $$f(X_t)=f(X_0)+int_0^tf'(X_s):{rm d}X_s+frac12int_0^tf''(X_s):{rm d}[X]_s;;;text{for all }tge0text{ almost surely}tag3.$$

Which assumption do we need to impose, if we want that $$left(int_0^tsigma_sf'(X_s):{rm d}W_sright)_{tge0}$$ is an $mathcal F$-martingale?

One possible assumption would be that $f$ has compact support and that $mu$ and $sigma$ are bounded on $left{(omega,t)inOmegatimes[0,infty):X_t(omega)inoperatorname{supp}fright}$.

I'm particularly interested in the case $mu_t=tildemu(t,X_t)$ and $sigma_t=tildesigma(t,X_t)$ for all $tge0$ for some Borel measurable $tildemu,tildesigma:[0,infty)timesmathbb Rtomathbb R$.

It is a martingale if the integrand is adapted and the expectation of the square of this expression (using ito's isometry) is finite. That is the general approach to such question.