Defining Correlated Multidimensional Brownian Motion

Summary

The document asks how to define a multidimensional Wiener process when its components are correlated. It proposes extending the one-dimensional conditions: the process starts at zero, has continuous paths, has independent increments, and each increment is multivariate normal with a covariance matrix that scales with elapsed time. The diagonal entries represent the variance of each component, while off-diagonal entries encode covariance between components.

It distinguishes this correlated process from a standard multidimensional Brownian motion with independent components. The discussion is a question and proposed definition rather than a sourced, confirmed treatment. In particular, the covariance matrix should be positive semidefinite; requiring it to be positive definite excludes valid cases with perfectly dependent components. Correlation of the component values at each time alone is not presented as sufficient to specify the joint process.

Key ideas

• A multidimensional Wiener process starts at the zero vector and has continuous paths.
• Its increments are independent across non-overlapping time intervals.
• Each increment is multivariate normal with covariance proportional to the elapsed time.
• Off-diagonal covariance terms describe dependence between components.
• A covariance matrix may be positive semidefinite, including degenerate dependence cases.

Tags

# Standard definition of multidimensional Brownian Motion with correlations


# Standard definition of multidimensional Brownian Motion with correlations












I was wondering was the standard definition of a multi-dimensional Brownian motion is. For one-dimension, I consider the following the standard definiton.

Brownian motion (or a Wiener process) is a real-valued stochastic process $W_{t}$ satisfying the following:

- $W(0)=0$ a.s.

- $W_{t}$ a.s.-continuous

- increments are independent

- $W_{t}-W_{s}\sim N(0,t-s)$, for $t>s$

I often find 2-dimensional Wiener process $(W_{1},W_{2})$ with correlation $\rho$ mentionned. And it is not quite clear how this is exactly defined. I assume that $W_{1}$ and $W_{2}$ being one-dimensional Wiener processes and $corr(W_{1}(t),W_{2}(t))=\rho$ is not enough, isn't it!? I would assume that we have to assume that $(W_{1}(t),W_{2}(t))$ has a two-dimensional normal distribution with mean 0.

Generally, is there a standard definition for a n-dimensional Wiener process with correlation? If so, I would be happy to get some references.

Else, my guess would be, similarly to the 1-dim case, that an n-dimensional Wiener process $W=(W_{1},\ldots,W_{n})$ is a n-dimensional process with:

- $W(0)=0$ a.s.

- $W_{t}$ a.s.-continuous

- increments are independent

- $W_{t}-W_{s}\sim N(0,\Sigma)$, for $t>s$

where

$\Sigma$ is a positive-definite and symmetric matrix with diagonal elements equal to $t-s$. The case $\Sigma=(t-s)\mathbf{1}$ would be a standard n-dimensional Wiener process, i.e. without any correlations.

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