Basic application of Ito's Lemma

Question

I'm just learning introductory stochastic calculus and I'm having a very difficult time figuring out how to apply Ito's Lemma. Suppose I want to find the semi-martingale decomposition of $e^tW_t$. I understand that, by Ito's lemma, this is $\int_0^te^sdW_s$ + $\int_0^tW_se^sds$. My question is how does one actually calculate the second term of ito's formula: $\int_0^tf'(X_s)dX_s$ + $\int_0^t(1/2)f''(X_s)d[X]_s$? I understand how the first term $\int_0^tf'(X_s)dX_s$ becomes $\int_0^te^sdW_s$, but, for example, in the second term, where does the $1/2$ go? Would someone be able to explain how we calculate this term? Of course also let me know if what I believe to be the answer is incorrect.

Answer 1

In the first formula you are using the Ito formula for non-autonomous transformations $y=f(t,x)$ with the increments $$ dY_t = f_t\,dt+f_x\,dX+\frac12f_{xx}\,d\langle X\rangle_t. $$ In the example used the function is linear in $x$, so that the last term does not occur. (Add the integral signs with $\int_0^tdY_s=Y_t-Y_0$ if the differential formalism does not seem to be exact enough.)

In the second formula you use an autonomous transformation $y=f(x)$ without independent time input. Thus the first term in the formula above becomes zero, $$ dY_t=f'(X_t)\,dX+\frac12f''(X_t)\,d⟨X⟩_t. $$

A combination of the example with the second case would be a constant multiple $y=cx$, which is not very interesting from the SDE perspective.