Exact Simulation of Jump-Diffusion Processes with Monte Carlo Applications

Abstract

We introduce a novel algorithm (JEA) to simulate exactly from a class of one-dimensional jump-diffusion processes with state-dependent intensity. The simulation of the continuous component builds on the recent Exact Algorithm ( (1)). The simulation of the jump component instead employes a thinning algorithm with stochastic acceptance probabilities in the spirit of (14). In turn JEA allows unbiased Monte Carlo simulation of a wide class of functionals of the process' trajectory, including discrete averages, max/min, crossing events, hitting times. Our numerical experiments show that the method outperforms Monte Carlo methods based on the Euler discretization.

On the exact and $\varepsilon$-strong simulation of (jump) diffusions

“…In this section, we present three novel Jump Exact Algorithms (JEAs). In contrast with existing algorithms [13,17,20], we note that the Bounded, Unbounded and Adaptive Unbounded Exact Algorithms in Section 3 can all be incorporated (with an appropriate choice of layered Brownian bridge construction) within any of the JEAs we develop. In Section 4.1, we present the Bounded Jump Exact Algorithm (BJEA), which is a reinterpretation and methodological extension of [13], addressing the case where there exists an explicit bound for the intensity of the jump process.…”

“…In contrast with existing algorithms [13,17,20], we note that the Bounded, Unbounded and Adaptive Unbounded Exact Algorithms in Section 3 can all be incorporated (with an appropriate choice of layered Brownian bridge construction) within any of the JEAs we develop. In Section 4.1, we present the Bounded Jump Exact Algorithm (BJEA), which is a reinterpretation and methodological extension of [13], addressing the case where there exists an explicit bound for the intensity of the jump process. In Section 4.2, we present the Unbounded Jump Exact Algorithm (UJEA) which is an extension to existing exact algorithms [17,20] in which the jump intensity is only locally bounded.…”

“…Recently, a new class of Exact Algorithms for simulating sample paths at finite collections of time points without approximation error have been developed for both diffusions [5,6,9,14] and jump diffusions [13,17,20]. These algorithms are based on rejection sampling, noting that sample paths can be drawn from the (target) measure T v 0,T by instead drawing sample paths from an equivalent proposal measure P v 0,T , and accepting or rejecting them with probability proportional to the Radon-Nikodým derivative of T v 0,T Figure 2.…”

“…In developing a methodological framework for simulating exact skeletons of (jump) diffusion sample paths we are able to present a number of significant extensions to the existing literature on exact algorithms [5,6,13,20]. In particular, we present novel exact algorithm constructions requiring the simulation of fewer points of the proposal sample path in order to evaluate whether to accept or reject (in effect a Rao-Blackwellisation of EA3 for diffusions [6], which we term the Unbounded Exact Algorithm (UEA), and a Rao-Blackwellisation of the Jump Exact Algorithms (JEAs) for jump diffusions [13,20], which we term the Bounded Jump Exact Algorithm (BJEA) and Unbounded Jump Exact Algorithm (UJEA), resp.) -all of which we recommend are adopted in practice instead of the existing equivalent exact algorithm.…”

Bayesian inference for hybrid discrete-continuous stochastic kinetic models

“…This allows efficient sampling of the slow reaction times via thinning, which is a point process variant of rejection sampling (Lewis & Shedler 1979). Related approaches have been proposed by Casella & Roberts (2011) and Rao & Teh (2013). The former consider simulation for jump-diffusion processes by combining a thinning algorithm with a generalisation of the exact algorithm (for diffusions) developed by Beskos & Roberts (2005), whilst the latter assume that an upper bound for the rate matrix governing the MJP is available and use uniformisation (Hobolth & Stone 2009) to simulate the process.…”