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 $X_2$?


1 Answer 1


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

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


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

  • $\begingroup$ I'm quite new to the theory. What is the name of the first equality and under which hypotheses is it true? $\endgroup$
    – Victor
    Commented Mar 31, 2019 at 20:00
  • 1
    $\begingroup$ @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 $X \sim \mathcal{N}(\mu, \sigma^2)$ $\endgroup$
    – rafaelc
    Commented Mar 31, 2019 at 20:06

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.