# Expectation in a stochastic differential equation

I'm new to stochastic calculus, I want to find the mean of $$X_2$$ with $$X_t = \exp(W_t)$$, with $$W_t$$ a Wiener process.

I used Ito's Lemma is arrive at the SDE: \begin{align} d(X_t) = \frac{1}{2}X_t dt + X_t dW_t \end{align} But how can I get the mean of $$X_2$$?

Assuming you are talking about unconditional expectation, in general you have

$$\mathbb{E}[X_t] = \mathbb{E}[e^{W_t}] = e^{\mathbb{E}[W_t] + \frac{1}{2}\text{Var}(W_t) }$$

which yields

$$\mathbb{E}[X_t]= e^{\frac{1}{2} t}$$

Hence,

$$\mathbb{E}[X_2]= e$$

• I'm quite new to the theory. What is the name of the first equality and under which hypotheses is it true? – Victor Mar 31 '19 at 20:00
• @Victor the first equality comes from the moment-generating function of a normal. Take a look here for more details. In general, $\mathbb{E}[e^X] = e^{\mu + \frac{1}{2} \sigma^2}$ holds whenever $X \sim \mathcal{N}(\mu, \sigma^2)$ – rafaelc Mar 31 '19 at 20:06