Expectation in a stochastic differential equation

Question

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 $X_2$?

rafaelc · Accepted Answer · 2019-03-31 19:52:27Z

3

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,

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

answered Mar 31, 2019 at 19:52

rafaelc

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$\begingroup$ 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

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