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.

No abstract

Abstract. This paper introduces a new class of numerical schemes for the pathwise approximation of solutions of stochastic differential equations (SDEs). The proposed family of strong predictor-corrector Euler methods are designed to handle scenario simulation of solutions of SDEs. It has the potential to overcome some of the numerical instabilities that are often experienced when using the explicit Euler method. This is of importance, for instance, in finance where martingale dynamics arise for solutions of SDEs with multiplicative diffusion coefficients. Numerical experiments demonstrate the improved asymptotic stability properties of the proposed symmetric predictor-corrector Euler methods.

In finance and economics the key dynamics are often specified via stochastic differential equations (SDEs) of jump-diffusion type. The class of jump-diffusion SDEs that admits explicit solutions is rather limited. Consequently, discrete time approximations are required. In this paper we give a survey of strong and weak numerical schemes for SDEs with jumps. Strong schemes provide pathwise approximations and therefore can be employed in scenario analysis, filtering or hedge simulation. Weak schemes are appropriate for problems such as derivative pricing or the evaluation of risk measures and expected utilities. Here only an approximation of the probability distribution of the jump-diffusion process is needed. As a framework for applications of these methods in finance and economics we use the benchmark approach. Strong approximation methods are illustrated by scenario simulations. Numerical results on the pricing of options on an index are presented using weak approximation methods. Copyright Springer Science+Business Media, LLC 2007Jump-diffusion processes, Discrete time approximation, Simulation, Strong convergence, Weak convergence, Benchmark approach, Growth Optimal portfolio, Primary 60H10, Secondary 65C05, G10, G13,

No abstract