Simulation of N-parameter Wiener Field

Question

In order to do some numerical experiment, I would like to simulate an N-parameter (or at least 2-parameter) Wiener Field (see : https://encyclopediaofmath.org/wiki/Wiener_field for definitions). I am sure this is pretty standard, but as I don't know much about stochastic simulations, I don't really see how to do it. Thus, if you know any references where this problem is tackled, that would allow me to fill my lack of knowledge !

Answer 1

A Wiener field is a particular Gaussian field and can be simulated in the same way as any Gaussian Field. To do this you utilise the defining property of a Gaussian field: If you evaluate a Gaussian field $G$ at $m$ points $x_1,\ldots,x_m$ the resulting $m$ random variables $G(x_1), \ldots,G(x_m)$ have a $m$-dimensional multivariate normal distribution. Their $m$-dimensional mean vector and their $m\text{-by-}m$ covariance matrix are obtained by evaluating the mean and covariance function of the field at the appropriate locations.

In your concrete case for $W$ this means:

1. Pick the points $x_1,\ldots,x_m$ in $[0,1]^n$ where you would like to simulate the field.
2. Since your process is zero mean you know that $W(x_1),\ldots,W(x_m)$ will be zero mean as well.
3. Calculate the covariance between the random variables at any two points $x_i$ and $x_j$ by plugging in the coordinates into the covariance function for the Wiener field: $$ \text{Cov}(x_i,x_j) = \prod_{d=1}^n\min(x_{i,d}, x_{j,d}).$$ This gives you the $m\text{-by-}m$ covariance matrix $C=\left( C_{i,j}\right)=\left(\text{Cov}(x_i,x_j)\right)$.
4. Finally put it all together by simulating multivariate normal vectors with zero mean and covariance $C$. Each simulated vector provides one instance of the values of the Wiener field at your chosen locations.