Optimal pointwise approximation of anticipating SDEs

Peter Parczewski

doi:10.3934/dcdsb.2023152

Article Contents

2024, Volume 29, Issue 4: 1730-1743. Doi: 10.3934/dcdsb.2023152

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Optimal pointwise approximation of anticipating SDEs

Peter Parczewski

Institute of Mathematics B6, 26, University of Mannheim, D-68131 Mannheim, Germany

Received: November 2022

Revised: July 2023

Early access: September 2023

Published: April 2024

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Abstract

We derive the optimal rate of convergence for the mean squared error at the terminal point for anticipating linear stochastic differential equations, where the integral is interpreted in Skorohod sense. Although alternative proof techniques are needed, our results can be seen as generalizations of the corresponding results for Itō SDEs. As a key tool we extend optimal approximation results for vectors of correlated Wiener integrals to general random vectors, which contain the solutions of our Skorohod SDEs.

Keywords:

Anticipating stochastic differential equations,
exact rate of convergence,
Malliavin calculus,
Skorohod integral,
Wiener chaos.

Mathematics Subject Classification: Primary: 60H07; Secondary: 65C30, 60H10.

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References

[1]	H. Ahn and A. Kohatsu-Higa, The Euler scheme for anticipating stochastic differential equations, Stochastics Stochastics Rep., 54 (1995), 247-269. doi: 10.1080/17442509508834008.
[2]	R. Buckdahn, Skorohod stochastic differential equations of diffusion type, Probab. Theory Related Fields, 93 (1992), 297-323. doi: 10.1007/BF01193054.
[3]	R. Buckdahn and D. Nualart, Linear stochastic differential equations and Wick products, Probab. Theory Related Fields, 99 (1994), 501-526. doi: 10.1007/BF01206230.
[4]	N. Chen and P. Glasserman, Malliavin Greeks without Malliavin calculus, Stochastic Process. Appl., 117 (2007), 1689-1723. doi: 10.1016/j.spa.2007.03.012.
[5]	L. Coutin, P. Friz and N. Victoir, Good rough path sequences and applications to anticipating stochastic calculus, Ann. Probab., 35 (2007), 1172-1193. doi: 10.1214/009117906000000827.
[6]	G. Di Nunno, B. Øksendal and F. Proske, Malliavin Calculus for Lévy Processes with Applications to Finance, Universitext, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-78572-9.
[7]	E. Fournié, J.-M. Lasry, J. Lebuchoux, P.-L. Lions and N. Touzi, Applications of Malliavin calculus to Monte Carlo methods in finance, Finance Stoch., 3 (1999), 391-412. doi: 10.1007/s007800050068.
[8]	C. Graham and D. Talay, Stochastic Simulation and Monte Carlo Methods, Éditions de l'école Polytechnique, Palaiseau, 2012.
[9]	H. Holden, B. Øksendal, J. Ubøe and T. Zhang, Stochastic Partial Differential Equations, A Modeling, White Noise Functional Approach, Second Edition, Springer, New York, 2010. doi: 10.1007/978-0-387-89488-1.
[10]	S. Janson, Gaussian Hilbert Spaces, Cambridge University Press, Cambridge, 1997. doi: 10.1017/CBO9780511526169.
[11]	P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Applications of Mathematics, 23. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.
[12]	H.-H. Kuo, White Noise Distribution Theory, Probability and Stochastics Series. CRC Press, Boca Raton, 1996.
[13]	T. Müller-Gronbach, Optimal pointwise approximation of SDEs based on Brownian motion at discrete points, Ann. Appl. Probab., 14 (2004), 1605-1642. doi: 10.1214/105051604000000954.
[14]	T. Müller-Gronbach and K. Ritter, Minimal errors for strong and weak approximation of stochastic differential equations, in Monte Carlo and Quasi-Monte Carlo Methods 2006, Springer, 2008, 53-82. doi: 10.1007/978-3-540-74496-2_4.
[15]	A. Neuenkirch and P. Parczewski, Optimal approximation of Skorohod integrals, J. Theor. Probab., 31 (2018), 206-231. doi: 10.1007/s10959-016-0716-2.
[16]	D. Nualart, The Malliavin Calculus and Related Topics, Second Edition. Probability and its Applications. Springer, New York, 2006.
[17]	P. Parczewski, A Wick functional limit theorem, Probab. Math. Statist., 34 (2014), 127-145.
[18]	É. Pardoux, Applications of anticipating stochastic calculus to stochastic differential equations, in Lecture Notes in Math., Springer-Verlag, Berlin, 1444 (1990), 63-105. doi: 10.1007/BFb0083610.
[19]	G. Shevchenko, Euler approximation for anticipating stochastic quasilinear differential equations, Theory Probab. Math. Statist., 72 (2006), 167-175. doi: 10.1090/S0094-9000-06-00674-0.
[20]	S. Torres and C. Tudor, The Euler scheme for a class of anticipating stochastic differential equations, Random Oper. Stochastic Equations, 12 (2004), 211-224. doi: 10.1515/1569397042222468.