Approximation of jump diffusions in finance and economics

Abstract

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,

Analysis of Explicit Tau-Leaping Schemes for Simulating Chemically Reacting Systems

“…Remark 2.1. Note that the coefficient c j (a; X) is not Lipschitz in X, which is different from the assumptions made in [5,6].…”

Almost sure exponential stability of numerical solutions for stochastic delay differential equations with jumps

“…The SDDEs with jumps (1.1) have been widely used in many branches of science and industry, in particular, in economics, finance and engineering (see, for example [1][2][3][4], and the references therein). Since most SDDEs with jumps cannot be solved explicitly, numerical methods have become essential.…”

Mean-square stability of the Euler–Maruyama method for stochastic differential delay equations with jumps

“…Recently, as one to build more realistic models, a class of SDEs with jumps, which are also referred to as jump-diffusion SDEs, have received much more attention. They have been used to describe the joint action of small, frequent transactions and rare, large movements of money in stochastic finance [3], and model population dynamics [1] (see also [17]), and jump-diffusion models have been applied in chemistry. Despite their wide interest, however, few analytical solutions have been proposed so far, thus, it is necessary to develop numerical methods and study the properties of these methods.…”