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.

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    $\begingroup$ 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. $\endgroup$ – andeliyeasi Apr 8 '14 at 12:34
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    $\begingroup$ correction, the relationship for more dimensions is $$ B_{\mathbf t} := W_{\mathbf t} - \prod_{j=1}^d t_j W_{(1,...,1)} $$ $\endgroup$ – andeliyeasi Apr 8 '14 at 12:46
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    $\begingroup$ Related: stats.stackexchange.com/questions/34354/… $\endgroup$ – David R Aug 26 '16 at 22:43

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