Approximation of quantiles of components of diffusion processes

In this paper we study the convergence rate of the numerical approximation of the quantiles of the marginal laws of ( X t ), where ( X t ) is a diffusion process, when one uses a Monte Carlo method combined with the Euler discretization scheme. Our convergence rate estimates are obtained under two s...

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Bibliographic Details
Published in:Stochastic processes and their applications 2004, Vol.109 (1), p.23-46
Main Authors: Talay, Denis, Zheng, Ziyu
Format: Article
Language:English
Subjects:
Euler method
Exact sciences and technology
Markov processes
Mathematics
Monte Carlo methods
Probability and statistics
Probability theory and stochastic processes
Sciences and techniques of general use
Simulation
Stochastic analysis
Stochastic differential equations
Stochastic differential equations Euler method Monte Carlo methods Simulation
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Summary:In this paper we study the convergence rate of the numerical approximation of the quantiles of the marginal laws of ( X t ), where ( X t ) is a diffusion process, when one uses a Monte Carlo method combined with the Euler discretization scheme. Our convergence rate estimates are obtained under two sets of hypotheses: either ( X t ) is uniformly hypoelliptic (in the sense of condition (UH) below), or the inverse of the Malliavin covariance of the marginal law under consideration satisfies condition (M) below. In order to deduce the required numerical parameters from our error estimates in view of a prescribed accuracy, one needs to get an as accurate as possible lower bound estimate for the density of the marginal law under consideration. This usually is a very hard task. Nevertheless, in our Section 3 of this paper, we treat a case coming from a financial application.
ISSN:0304-4149
1879-209X
DOI:10.1016/j.spa.2003.06.001