Error Analysis for Approximation of Stochastic Differential Equations Driven by Poisson Random Measures
Abstract
Let Xt be the solution of a stochastic differential equation (SDE) with starting point x0 driven by a Poisson random measure. Additive functionals are of interest in various applications. Nevertheless they are often unknown and can only be found by simulation on computers. We investigate the quality of the Euler approximation. Our main emphasis is on SDEs driven by an $\alpha$-stable process, $0<\alpha < 2$, where we study the approximation of the Monte Carlo error ${\Bbb E} [f(X_T)]$, f belonging to ${L}^\infty$. Moreover, we treat the case where the time equals $T\wedge \tau$, where $\tau$ is the first exit time of some interval.
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References
1.
V. Bally, D. Talay, The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function, Probab. Theory Related Fields, 104 (1996), 43–60
2.
Vlad Bally, Denis Talay, The law of the Euler scheme for stochastic differential equations. II. Convergence rate of the density, Monte Carlo Methods Appl., 2 (1996), 93–128
3.
R. Bass, M. Cranston, The Malliavin calculus for pure jump processes and applications to local time, Ann. Probab., 14 (1986), 490–532
4.
Jean Bertoin, Lévy processes, Cambridge Tracts in Mathematics, Vol. 121, Cambridge University Press, 1996x+265
5.
Klaus Bichteler, Jean‐Bernard Gravereaux, Jean Jacod, Malliavin calculus for processes with jumps, Stochastics Monographs, Vol. 2, Gordon and Breach Science Publishers, 1987x+161
6.
Patrick Billingsley, Convergence of probability measures, John Wiley & Sons Inc., 1968xii+253
7.
E. Çinlar, J. Jacod, Representation of semimartingale Markov processes in terms of Wiener processes and Poisson random measures, Progr. Prob. Statist., Vol. 1, Birkhäuser Boston, Mass., 1981, 159–242
8.
E. Çinlar, J. Jacod, P. Protter, M. Sharpe, Semimartingales and Markov processes, Z. Wahrsch. Verw. Gebiete, 54 (1980), 161–219
9.
. Gīhman, A. Skorohod, Stochastic differential equations, Springer‐Verlag, 1972viii+354, Translated from the Russian by Kenneth Wickwire; Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 72
10.
Emmanuel Gobet, Weak approximation of killed diffusion using Euler schemes, Stochastic Process. Appl., 87 (2000), 167–197
11.
M. Barlow, S. Jacka, M. Yor, Inequalities for a pair of processes stopped at a random time, Proc. London Math. Soc. (3), 52 (1986), 142–172
12.
Shūya Kanagawa, The rate of convergence for approximate solutions of stochastic differential equations, Tokyo J. Math., 12 (1989), 33–48
13.
A. Kohatsu‐Higa, P. Protter, The Euler scheme for SDE’s driven by semimartingales, Pitman Res. Notes Math. Ser., Vol. 310, Longman Sci. Tech., Harlow, 1994, 141–151
14.
Thomas Kurtz, Philip Protter, Wong‐Zakai corrections, random evolutions, and simulation schemes for SDEs, Academic Press, Boston, MA, 1991, 331–346
15.
Philip Protter, Stochastic integration and differential equations, Applications of Mathematics (New York), Vol. 21, Springer‐Verlag, 1990x+302, A new approach
16.
Philip Protter, Denis Talay, The Euler scheme for Lévy driven stochastic differential equations, Ann. Probab., 25 (1997), 393–423
17.
Denis Talay, Probabilistic numerical methods for partial differential equations: elements of analysis, Lecture Notes in Math., Vol. 1627, Springer, Berlin, 1996, 148–196
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Published In
SIAM Journal on Numerical Analysis
Pages: 87 - 113
ISSN (online): 1095-7170
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Copyright © 2002 Society for Industrial and Applied Mathematics.
History
Published online: 26 July 2006
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