# Calculation of a process's drift

Let $$X_t:=e^{W_t}$$ where $$W_t$$ follows the Wiener process. Calculate the drift.

The answer is given as $$X_t/2$$. My attempt at a solution (which I'm afraid is poor from a mathematical standpoint):

I applied Ito's lemma as $$dX_t=\frac{\partial X_t}{\partial W_t}dW_t+\frac{1}{2}\frac{\partial^2 X_t}{\partial W_t^2}(dW_t)^2$$ and using the fact that $$(dW_t)^2=dt$$, we get: $$dX_t=\frac{e^{W_t}}{2}dt+e^{W_t}dW_t$$ Therefore the drift is indeed $$X_t/2$$.

Is my derivation correct? I would appreciate any input on that.

Your solution is correct. Generally speaking, for any $$\alpha,\beta\in\mathbb{R}$$, the drift $$\mu_{X^{\alpha\beta}}$$ of the process: $$X_t^{\alpha\beta}:=e^{\alpha t+\beta W_t}$$ will be equal to: $$\mu_{X^{\alpha\beta}}=\left(\alpha+\frac{\beta^2}{2}\right)X_t^{\alpha\beta}$$