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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:

  1. $W(0)=0$ a.s.
  2. $W_{t}$ a.s.-continuous
  3. increments are independent
  4. $W_{t}-W_{s}\sim N(0,t-s)$, 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:

  1. $W(0)=0$ a.s.
  2. $W_{t}$ a.s.-continuous
  3. increments are independent
  4. $W_{t}-W_{s}\sim N(0,\Sigma)$, 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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  • $\begingroup$ From most books I saw, a multi-dimensional Brownian motion has independent components, which is the reason that I assumed component independence when I first answered this question. For correlated Brownian motions with a given co-variance matrix, we can call it a vector of correlated Brownian motions. $\endgroup$ – Gordon Sep 16 '17 at 14:12

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