A collection of numerical codes to help perform simulations of ambit fields.

We are working towards providing a library of useful C-codes. Currently, the project is very much in its infancy as we experiment and explore.

• Use GSL_RND_SEED to properly propagate seeds of the random number generators.

• The documentation is still somewhat incomplete.

The directory ambit-stochastics/software contains a few executables which are described below.

To see a short summary of the use of the program, try

./simulate-trawl-process --help

With the --cascade option, the program simulates from the process X(t) = L(A + (0,t)) where L is a homogeneous Levy basis and A is an ambit set given by

A = {(x,t) | 0 <= t <= T and |x| <= g(t)},

where

g(t) = ((1 - (t / T) ^ s) / (1 + (L t / T) ^ s)) ^ (1 / s).

Then the process Y = exp(X) displays scaling of correlators when T/L << lag << T. The parameter s determines the smoothness of the correlator when the lag is near T and T/L.

The Levy basis can be either normal (specified with the --normal option) or generalised hyperbolic (specified with the --generalised-hyperbolic option). The parameters of the corresponding Levy seed are lambda, alpha, beta, mu, and delta in the generalised hyperbolic case. In the normal case, the mean and variance are given by mu and delta, respectively.

The time step of the simulation is specified using the --resolution option. The option --samples specifies the length of the simulation, i.e., the number of samples to generate. The output is stored as the dataset /simulation in the HDF5 file specified by --output. The output file is overwritten if it already exists.

Suppose we want to generate one million samples from the process X such that the process Y = exp(X) has the following properties.

• One million samples are generated.
• The output is stored in $HOME/output.h5.
• The decorrelation length T is 1.
• The cascade length L is 1000.
• The smoothness parameter s is 2.
• A normal Levy basis is used.
• The mean of Y is 1.
• The correlator of exponent tau_11 of order (1,1) of Y is 0.1.
• The time step is 0.005.

First, we calculate that the parameters mu and delta of the Levy basis must be

delta = 1 / 2 * tau_11 * L / T =  50,
mu    = -1 / 2 * delta         = -25.

Next, we run the simulation.

./simulate-trawl-process --samples=1000000 --output=$HOME/output.h5 \
    --cascade --decorrelation-length=1 --cascade-length=1000 \
    --smoothness=2 --normal --mu=-25 --delta=50 --resolution=0.005

The simulation takes a few seconds to complete.

Suppose now that we want to do a similar simulation but with a normal inverse Gaussian Levy basis with shape parameters alpha = 2.25 and beta = -1.75. Let gamma(p) = sqrt(alpha^2 - (beta + p)^2). Then

delta = 1 / 2 * L / T * tau_11 / (2 * gamma(1) - gamma(0) - gamma(2)) = 84.41,
mu    = -delta * (gamma(0) - gamma(1))                                = 59.69

Next, we run the simulation.

./simulate-trawl-process --samples=1000000 --output=$HOME/output.h5 \
    --cascade --decorrelation-length=1 --cascade-length=1000 \
    --smoothness=2 --generalised-hyperbolic --lambda=-0.5 --alpha=2.25 \
    --beta=-1.75 --mu=59.69 --delta=84.41 --resolution=0.005

It also takes a few seconds to complete, though a bit longer than when the normal Levy basis is used.

Other shapes of ambit sets are supported, but the documentation on how to use this feature still has to be written.

Sample from the generalised inverse Gaussian distribution. The library supports the boundary cases of the parameter region.

Sample from multivariate and univariate generalised hyperbolic distributions. The library supports the boundary cases of the parameter region.

Sample from multivariate and univariate normal distributions

Generate time series from trawl processes.

Various small functions used by the other libraries.