Coupled Brownian Motion

摘要

We present a way of considering a stochastic process {B_t:t≥0} with values in R~2 such that each component is a Brownian motion. The distribution function of B_t, for each t, is obtained as the copula of the distribution functions of the components. In this way a "coupled Brownian motion" is obtained. The (one-dimensional) Brownian motion is the example of a stochastic process that (a) is a Markov process, (b) is a martingale in continuous time, and (c) is a Gaussian process. It will be seen that while the coupled Brownian motion is still a Markov process and a martingale, it is not in general a Gaussian process.