This is probably an easy question but I am getting aquanted with Ito's formula and stuck on an exercise in my textbook.
Let $X_{t}=W_{t}-a t/2$ where $a$ is a real number and $W_{t}$ is brownian motion with $W_0=0$
I want to use the general Ito formula but unsure how to treat the derivatives when having both $W_{t}$ and $t$. Any advise in the right direction would be much appreciated.
This is what I mean by Ito's general formula:
$f(X) = f(X_0)+\int f^\prime(X)\,dX + \frac{1}{2}\int f^{\prime\prime}(X)\,d[X]$
We have $dX_t = dW_t - \frac{a}{2}dt$ and $d[X]_t = d[W]_t = dt$ since the deterministic piece $-at/2$ doesn't contribute to the quadratic variation. So, $$ f(X_t) = f(X_0)+\int_0^t f'(X_s)\, dW_s - \frac{a}{2} \int_0^t f'(X_s) \,ds + \frac{1}{2} \int_0^t f''(X_s) \, ds \\ = f(X_0)+\int_0^t f'(X_s)\, dW_s + \frac{1}{2} \int_0^t \big[f''(X_s) - af'(X_s) \big]\,ds $$