Finding the Mean of an Exponential Wiener Process

Summary

The document shows how to calculate the unconditional expectation of an exponential Wiener process. For a process defined as the exponential of standard Brownian motion, the answer uses the moment-generating property of a normal random variable: the expectation of the exponential depends on both the process mean and half its variance. Since Brownian motion has zero mean and variance equal to elapsed time, this gives an expectation that grows exponentially with time.

The response evaluates the result at the requested time and clarifies that it assumes an unconditional expectation. It does not derive the result from the displayed Itô differential equation, discuss conditional expectations, or address extensions such as drifted Brownian motion. Its value is a compact application of the lognormal moment formula rather than a trading strategy or empirical market analysis.

Key ideas

• The exponential of a normally distributed variable has an expectation determined by its mean and variance.
• Standard Brownian motion has zero mean and variance equal to elapsed time.
• The unconditional mean of the exponential Wiener process grows as the exponential of half the elapsed time.
• The calculation assumes standard Brownian motion and an unconditional expectation.

Tags

# Expectation in a stochastic differential equation


# 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 $X_2$?

## Answer by rafaelc (score 3, accepted)

https://quant.stackexchange.com/a/44856

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,

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

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