Numerical aspects of shot noise representation of infinitely divisible laws and related processes

Abstract

The ever-growing appearance of infinitely divisible laws and related processes in various areas, such as physics, mathematical biology, finance and economics, has fuelled an increasing demand for numerical methods of sampling and sample path generation. In this survey, we review shot noise representation with a view towards sampling infinitely divisible laws and generating sample paths of related processes. In contrast to many conventional methods, the shot noise approach remains practical even in the multidimensional setting. We provide a brief introduction to shot noise representations of infinitely divisible laws and related processes, and discuss the truncation of such series representations towards the simulation of infinitely divisible random vectors, Lévy processes, infinitely divisible processes and fields and Lévy-driven stochastic differential equations. Essential notions and results towards practical implementation are outlined, and summaries of simulation recipes are provided throughout along with numerical illustrations. Some future research directions are highlighted.

A general approach to sample path generation of infinitely divisible processes via shot noise representation

“…In this paper, we develop the general framework of simulating infinitely divisible processes driven by Lévy processes without Gaussian components based on their shot noise representations. The proposed framework unifies and generalizes the existing case-by-case approaches, as summarized in [22], and offers a sample path generation scheme that is indeed a lot easier than simulation of general Gaussian processes. We provide technical conditions for error analysis in terms of the integrator and the deterministic kernel and include an extensive collection of infinitely divisible processes of interest in the literature so as to exemplify the effectiveness of the proposed approach.…”

“…where {L s : s ∈ T } is the Lévy process characterized by Lévy-Khintchine triple (0, 0, ν) over [0, T ], {Γ k } k∈N are the arrival times of the standard Poisson process, {U k } k∈N are iid copies of U, {T k } k∈N are iid uniform random variables over T , with mutual independence of the random sequences and {c k } k∈N is a sequence of centers in R d with c k := k k−1 E[H(s,U)½ (0,1] ( H(s,U) )] ds for k ∈ N. We remark that the decomposition (2.4) exists in a similar form to the so-called the inverse Lévy measure method [7], whereas the resulting expression often does not offer a viable numerical method [11]. In fact, the decomposition (2.4) is not unique but can be employed to derive a few distinct shot noise representations of an infinitely divisible random vector via, not only the aforementioned inverse Lévy measure method but also, the rejection, thinning and Bondesson's methods [22]. Thanks to the non-uniqueness, each of the three typical Lévy measures (Section 4) admits at least one implementable expression for simulation purposes [10,14].…”

Approximation and Error Analysis of Forward-Backward SDEs driven by General Lévy Processes using Shot Noise Series Representations

“…We next mention some important facts on Lévy processes which we need to approximate the infinite Lévy measure. We opt for the approximation using series representations which goes back to Rosiński (2001), see Yuan & Kawai (2021) for a recent overview. Let L be a pure jump Lévy process on (Ω, F, (F t ) 0≤t≤T , P) with Lévy measure ν as discussed above.…”

“…The most convenient representation is case-dependent given the specific Lévy measure. Well-known special cases include the inverse Lévy measure method, the rejection method or the thinning method, see Yuan & Kawai (2021) for details. To obtain a feasible numerical algorithm one has to truncate the infinite series in (6).…”

“…We now discuss an alternative where we approximate the small jumps by a scaled Brownian motion and discuss the approximation error. The idea of approximating small jumps goes back to Asmussen & Rosiński (2001), for a gentle introduction see Yuan & Kawai (2021). Note that this approach is not generally valid for every Lévy process and requires additional assumptions given below.…”