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Approximation of quantiles of components of diffusion processes

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  • Talay, Denis
  • Zheng, Ziyu

Abstract

In this paper we study the convergence rate of the numerical approximation of the quantiles of the marginal laws of (Xt), where (Xt) 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 (Xt) 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.

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Article provided by Elsevier in its journal Stochastic Processes and their Applications.

Volume (Year): 109 (2004)
Issue (Month): 1 (January)
Pages: 23-46

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Handle: RePEc:eee:spapps:v:109:y:2004:i:1:p:23-46

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Keywords: Stochastic differential equations Euler method Monte Carlo methods Simulation;

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  1. Denis Talay & Ziyu Zheng, 2003. "Quantiles of the Euler Scheme for Diffusion Processes and Financial Applications," Mathematical Finance, Wiley Blackwell, vol. 13(1), pages 187-199.
  2. Arturo Kohatsu & Roger Pettersson, 2002. "Variance reduction methods for simulation of densities on Wiener space," Economics Working Papers 597, Department of Economics and Business, Universitat Pompeu Fabra.
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