Some Monte Carlo methods for jump diffusions

In this thesis we develop computationally efficient methods to simulate finite dimensional representations of (jump) diffusion and (jump) diffusion bridge sample paths over finite intervals, without discretisation error (exactly), in such a way that the sample path can be restored at any desired finite collection of time points. Furthermore, we extend methodology for particle filters to the setting in which the transition density of the latent process is governed by a jump diffusion. Finally, we present methodology which allows the simulation of upper and lower bounding processes which almost surely constrain (jump) diffusion and (jump) diffusion bridge sample paths to any specified tolerance. We demonstrate the efficacy of our approach by showing that with finite computation it is possible to determine whether or not sample paths cross various irregular barriers, simulate to any specified tolerance the first hitting time of the irregular barrier, and simulate killed diffusion sample paths.

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