Numerical Solution of Stochastic Differential Equations with Jumps in Finance

Abstract

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblikPrinted on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) PrefaceThis research monograph concerns the design and analysis of discrete-time approximations for stochastic differential equations (SDEs) driven by Wiener processes and Poisson processes or Poisson jump measures. In financial and actuarial modeling and other areas of application, such jump diffusions are often used to describe the dynamics of various state variables. In finance these may represent, for instance, asset prices, credit ratings, stock indices, interest rates, exchange rates or commodity prices. The jump component can capture event-driven uncertainties, such as corporate defaults, operational failures or insured events. The book focuses on efficient and numerically stable strong and weak discrete-time approximations of solutions of SDEs. Strong approximations provide efficient tools for simulation problems such as those arising in filtering, scenario analysis and hedge simulation. Weak approximations, on the other hand, are useful for handling problems via Monte Carlo simulation such as the evaluation of moments, derivative pricing, and the computation of risk measures and expected utilities. The discrete-time approximations considered are divided into regular and jump-adapted schemes. Regular schemes employ time discretizations that do not include the jump times of the Poisson jump measure. Jump-adapted time discretizations, on the other hand, include these jump times.The first part of the book provides a theoretical basis for working with SDEs and stochastic processes with jumps motivated by applications in finance. This part also introduces stochastic expansions for jump diffusions. It further proves powerful results on moment estimates of multiple stochastic integrals. The second part presents strong discrete-time approximations of SDEs with given strong order of convergence, including derivative-free and predictor-corrector schemes. The strong convergence of higher order schemes for pure jump SDEs is established under conditions weaker than those required for jump diffusions. Estimation and filtering methods are discussed. The third part of the book introduces a range of weak approximations with jumps. These weak approximations include derivative-free, predictor-corrector, and VI Preface simplified schemes. The final part of the research monograph raises questions on numerical stability and discusses powerful martingale representations and variance reduction techniques in the context of derivative pricing.

Second-order schemes for solving decoupled forward backward stochastic differential equations

“…Here we briefly recall some classic numerical schemes and their properties discussed in [25]. Any of these schemes can serve as the approximation of the forward SDE in our schemes for the FBSDE.…”

“…By using the Itô-Taylor expansion, numerical schemes of strong-order β (or the weak-order β) [25] can be represented in a general form, i.e., X n+1 =X n + Φ(t n , t n+1 , X n , I J ∈A β ), (3.4) where Φ is the incremental, A β is a hierarchical set such that the convergence rate of the scheme is β in a strong or weak sense. Details of the index set I J ∈A β and the definition of A β can be found in [25] (pp. 196 and pp.…”

“…According to Theorem 6.4.1 and Theorem 12.3.4 in [25], it is easy to derive that α = β = γ for strong and weak Taylor schemes. In this paper, we prove in Theorem 4.2 that the secondorder convergence of the proposed semi-discrete scheme for the FBSDE in (2.1) requires α = β = γ = 2.…”

“…In finance and insurance, FBSDEs-based approaches [23,29] have gained a great attention by both academics and practitioners, because FBSDEs provide us a unified framework to describe the mathematical problems which arise in option pricing [13], portfolio hedging [14], market utility maximization [2] and risk measures [24,26], etc. Moreover, in the presence of jump behaviors in many financial problems [25], Lèvy jump processes have been incorporated into FBSDEs [11,14], so as to accurately capture and properly interpret event-driven stochastic phenomena, such as corporate defaults, operational failures, insured events, etc. In mathematics, one can relate FBSDEs with jumps to a class of nonlinear partial integro-differential equations (PIDEs), based on the extension of the nonlinear Feynman-Kac theory studied in [1].…”

“…A semi-discrete scheme is developed for discrete time approximation, which is constituted by a classic scheme for the forward SDE [17,25] and a novel scheme for the backward SDE. Under some reasonable regularity conditions, we prove that the semi-discrete scheme can achieve second-order convergence in approximating the FBSDEs of interest; and such convergence rate does not require jump-adapted temporal discretization.…”