  Generalised hyperbolic (GH) processes are a class of stochastic processes
that are used to model the dynamics of a wide range of complex systems that
exhibit heavy-tailed behavior, including systems in finance, economics,
biology, and physics. In this paper, we present novel simulation methods based
on subordination with a generalised inverse Gaussian (GIG) process and using a
generalised shot-noise representation that involves random thinning of infinite
series of decreasing jump sizes. Compared with our previous work on GIG
processes, we provide tighter bounds for the construction of rejection sampling
ratios, leading to improved acceptance probabilities in simulation.
Furthermore, we derive methods for the adaptive determination of the number of
points required in the associated random series using concentration
inequalities. Residual small jumps are then approximated using an appropriately
scaled Brownian motion term with drift. Finally the rejection sampling steps
are made significantly more computationally efficient through the use of
squeezing functions based on lower and upper bounds on the L\'evy density.
Experimental results are presented illustrating the strong performance under
various parameter settings and comparing the marginal distribution of the GH
paths with exact simulations of GH random variates. The new simulation
methodology is made available to researchers through the publication of a
Python code repository.
