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Mean and Variance of a Complex Brownian Itô Integral

Article Quant Q&A · Author: Thomas J
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Summary

The document considers the stochastic integral of the complex exponential of standard Brownian motion against that same Brownian motion. It identifies the integral as a martingale with zero expectation, then derives its variance using Itô’s isometry. The remaining expectation inside the variance integral is evaluated with the moment-generating function of a centered normal variable, giving a closed-form expression that depends on time.

The explanation is concise and relies on standard stochastic-calculus results. It does not discuss broader applications, assumptions needed for the isometry, or alternative derivations. The variance calculation treats the complex integral through the stated isometry expression, so readers applying it should be mindful of the convention used for variance of a complex-valued random variable.

Key ideas

  • The stochastic integral is presented as a martingale with zero mean.
  • Itô’s isometry converts the second-moment calculation into an integral over time.
  • The normal moment-generating function evaluates the complex exponential expectation.
  • The resulting variance increases with time toward a finite limit.

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# How to find the mean and variance of this stochastic process?


# How to find the mean and variance of this stochastic process?












$ I_t = \int_0^t e^{i W_s} dWs $ where $W_s$ is the standard brownian motion and $i$ is the complex number. Any help will be appreciated!

## Answer by Pandaaaaaaa (score 1)

https://quant.stackexchange.com/a/32512

This process is martingale and we have

$$ E[I_t|t=0]=0 $$

To find the variance, let's write it into differential form $$ dI_t =e^{iW_t}dW_t $$ Apply Ito's isometry $$ Var(I_t)=\int_0^tE[e^{2iW_s}]ds $$ Apply MGF of normal $$ Var(I_t)=\int_0^te^{\frac{1}{2}(2i)^2s}ds=\int_0^te^{-2s}ds=\frac{1-e^{-2t}}{2} $$ Please let me know if anything is incorrect.

## Answer by Saïd Naciri (score -2)

https://quant.stackexchange.com/a/32551

The expectation of an Itô integral (and a Wiener integral) is always zero.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.