Let $f$ be a real-valued function with bounded continuous second derivative $f''$, and $w(t)$ be a Wiener process. Let $$ V(t,w(t)) = f(w(t)) - \frac{1}{2} \int_a^t f''(w(s))ds. $$
I want to apply I want to apply the following formula
$$dV(t,w(t)) = \left(\frac{\partial}{\partial t}V(t,w(t)) + \frac{1}{2} \frac{\partial^2}{\partial x^2}V(t,w(t))\right)dt + \frac{\partial}{\partial x}V(t,w(t))dw(t)$$ to find the differential of $V(t,w(t))$.