# How to create a multivariate Brownian Bridge

It is known, that a standard multivariate Brownian bridge $y(\mathbf u)$ is a centered Gaussian process with covariance function $$\mathbb E(y(\mathbf u) y(\mathbf v)) = \prod_{j=1}^d (u_j \wedge v_j) - \prod_{j=1}^d u_j v_j$$

I am not sure about how to constuct such a multivariate Brownian bridge.

My first thought was to start somehow with a univariate Brownian bridge. I have found information about that and even a package in R that can do this, but only for the univariate Brownian bridge.

I found this, but as I understand it, what has been done there is not a standard multivariate Brownian bridge as defined above or e.g. in this paper.

I would appreciate any hints and support.

• As I found out in Deheuvels paper link there is the following relationship between a Brownian Bridge $B_t$ and a Brownian Sheet (or Wiener Sheet) $W_t$: $$B_t := W_t - \frac t T W_T$$ So I think the problem reduces to simulating a Brownian sheet. I will ask my questions about this in a seperate question. – andeliyeasi Apr 8 '14 at 12:34
• correction, the relationship for more dimensions is $$B_{\mathbf t} := W_{\mathbf t} - \prod_{j=1}^d t_j W_{(1,...,1)}$$ – andeliyeasi Apr 8 '14 at 12:46
• – David R Aug 26 '16 at 22:43