Why a Complex Brownian Diffusion Has Positive Variance

Summary

The document asks whether the process driven by an imaginary multiple of real Brownian motion has negative variance. The answer distinguishes the process’s imaginary-valued increments from its variance: at time t, the process is an imaginary multiple of Brownian motion, and its variance is σ²t, which is nonnegative.

It points to the usual definition of variance for complex random variables, under which variance is real and nonnegative and reflects the variances of the real and imaginary components. The exchange gives a concise conceptual correction, but does not derive the result or discuss alternative conventions such as complex quadratic variation. That distinction matters because the title’s reference to negative quadratic variation is not addressed directly; the explanation concerns variance of the process at a fixed time.

Key ideas

• Multiplying real Brownian motion by the imaginary unit does not make its variance negative.
• The stated variance at time t is σ²t.
• Complex random variable variance is real and nonnegative under the convention described.
• Variance and quadratic variation are distinct concepts, and the answer focuses on variance.

Tags

# Process with negative quadratic variation


# Process with negative quadratic variation












Today seems to be question day for me, sorry.

The complex process

$$ dX = i\sigma dW $$

where $i = \sqrt{-1}$ and $dW$ is a standard (real-valued) Brownian motion will have a negative variance correct?

## Answer by Valometrics.com (score 2)

https://quant.stackexchange.com/a/50818

$i \times \sigma \times W$ is a solution of your equation. Its variance at time $t$ is equal to $\sigma^2 \times t$ which is positive.

Please check this page for more details about how to compute variance for complex random variables:

Wikipedia: complex random variables

> The variance is always a nonnegative real number. It is equal to the sum of the variances of the real and imaginary part of the complex random variable

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