Localization and Exact Simulation of Brownian Motion-Driven Stochastic Differential Equations

Abstract

Generating sample paths of stochastic differential equations (SDE) using the Monte Carlo method finds wide applications in financial engineering. Discretization is a popular approximate approach to generating those paths: it is easy to implement but prone to simulation bias. This article presents a new simulation scheme to exactly generate samples for SDEs. The key observation is that the law of a general SDE can be decomposed into a product of the law of standard Brownian motion and the law of a doubly stochastic Poisson process. An acceptance-rejection algorithm is devised based on the combination of this decomposition and a localization technique. The numerical results corroborates that the mean-square error of the proposed method is in the order of O(t −1/2), which is superior to the conventional discretization schemes. Furthermore, the proposed method also can generate exact samples for SDE with boundaries which the discretization schemes usually find difficulty in dealing with.

Functionals of Multidimensional Diffusions with Applications to Finance

“…The aim of this section is to introduce two alternatives to discretization schemes, which allow us to eliminate the discretization error introduced by discretization schemes and to recover the Monte Carlo convergence rate. These alternatives are the exact simulation methods, due to Roberts and collaborators, see Beskos et al (2006), Beskos and Roberts (2005), and also Chen and Huang (2012b), and multilevel methods due to coauthors, see Giles (2008a, 2008b). We firstly provide a very brief introduction to Monte Carlo methods and then briefly illustrate the Euler discretization scheme, which motivates the exact simulation and multilevel methods.…”

Monte Carlo Algorithms for Default Timing Problems

“…If these samples are exact, then so are the samples of the T n . The methods of Beskos and Roberts (2005) and Chen (2009) can be used to generate exact samples of a broad range of diffusion processes n . The method of Giesecke and Smelov (2010) supplies exact samples of jump-diffusion processes.…”

“…This inequality facilitates an acceptance-rejection algorithm to draw a sample of . A sample X from g · y is accepted as a sample of with probability p X y H ; X can be generated by the inverse method or the exact methods of Beskos and Roberts (2005) and Chen (2009). By the continuity and Markov property of the process ,…”

Finite variance unbiased estimation of stochastic differential equations

“…Some of the recent approaches in this direction involve generating unbiased samples of g(X t,x T ) using the method of exact simulation by Beskos and Roberts (2005) and Chen and Huang (2013). Note that these schemes essentially handle one-dimensional situations.…”