Suppose I have a stochastic process $X_t$ that satisfies the SDE:
$$dX_{t}=\mu(X_{t})dt+\sigma(X_{t})dW_{t}$$
where $W_t$ is a Brownian motion. Suppose I haven't made any assumption yet about the functions $\sigma(\cdot)$ and $\mu(\cdot)$ (therefore I don't even know if my SDE has a strong solution, or even a weak one).
Can I use Ito's formula anyway? In other words, is it true that the process $Y_t=f(X_t)$, where $f(\cdot)$ is twice differentiable, follows an SDE
$$dY_t=\mu(X_t)f'(X_t)dt+\frac{1}{2}f''(X_t)\sigma(X_t)dW_t$$
I couldn't find a simple answer in the usual references, any help is welcome.
Best Answer
Implicit in your initial supposition "Suppose I have a stochastic process $X_t$ that satisfies the SDE:..." is the hypothesis that the integrals $\int_0^t\mu(X_s)\,ds$ and $\int_0^t\sigma(X_s)\,dW_s$ are well defined for all $t>0$, almost surely; namely, that $\mu$ and $\sigma$ are Borel-measurable functions such $\int_0^t|\mu(X_s)|\,ds<\infty$ and $\int_0^t|\sigma(X_s)|^2\,ds<\infty$ for all $t>0$, almost surely. If this is so then Ito's formula can be used for a $C^2$ function $f$.