We present a way of consodering a stochastic process $\{B_t:t\ge 0\} with values in $\mathbb{R}^2$ such 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 nthis way "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.
Coupled Brownian Motion
SEMPI, Carlo
2010-01-01
Abstract
We present a way of consodering a stochastic process $\{B_t:t\ge 0\} with values in $\mathbb{R}^2$ such 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 nthis way "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.File in questo prodotto:
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