Using Lévy’s Characterization to Identify an Itô Integral as Brownian Motion

Summary

The document asks for the distribution and exponential moment of an Itô integral whose integrand is the sign of a Brownian motion. The answer observes that the integral is a continuous, square-integrable martingale and computes its quadratic variation from the squared sign function. Since that integrand has squared magnitude one, the quadratic variation grows linearly with time.

Lévy’s characterization then identifies the integral itself as a standard Brownian motion. Its distribution and exponential moment follow from standard Brownian motion results. The explanation is concise and hinges on the martingale and quadratic-variation conditions; it does not provide the resulting formulas or discuss broader applications. This is a probability result rather than a trading strategy, though it may be useful background for quantitative finance involving stochastic processes.

Key ideas

• The integral of the Brownian sign process is a continuous, square-integrable martingale.
• Its quadratic variation equals elapsed time because the squared sign integrand is one.
• Lévy’s characterization identifies the integral as standard Brownian motion.
• Its distribution and exponential moment therefore follow from the standard Brownian law.

Tags

# Compute distribution of a stochastic variable


# Compute distribution of a stochastic variable












$sign(x)=1$ if $x\geq0$

$sign(x)=-1$ if $x< 0$

Consider $$ X_t = \int^t_0 sign(W_u)dW_u $$ where $W_t$ is a wiener proces.

How can I determine the distribution of $X_t$ and compute $E[\exp(\lambda X_t )]$?

## Answer by Gordon (score 5, accepted)

https://quant.stackexchange.com/a/43192

Note that $\{X_t, \, t \ge 0\}$ is continuous, square-integrable martingale with quadratic variation process \begin{align*} \langle X\rangle_t = \int_0^t {\rm sign}^2(W_s)\, ds =t. \end{align*} Then, it is a standard Brownian motion based on Levy’s Characterization of Brownian Motion. The remaining is straightforward.

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