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Discrete-Time Approximation of Decoupled Forward‒Backward Stochastic Differential Equations Driven by Pure Jump Lévy Processes

Part of: Stochastic analysis Markov processes

Published online by Cambridge University Press:  04 January 2016

Soufiane Aazizi
Soufiane Aazizi*
Affiliation:
Cadi Ayyad University
*
∗ Postal address: Department of Mathematics, Faculty of Sciences Semlalia Cadi Ayyad University, B.P. 2390 Marrakesh, Morocco. Email address: aazizi.soufiane@gmail.com
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Abstract

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We present a new algorithm to discretize a decoupled forward‒backward stochastic differential equation driven by a pure jump Lévy process (FBSDEL for short). The method consists of two steps. In the first step we approximate the FBSDEL by a forward‒backward stochastic differential equation driven by a Brownian motion and Poisson process (FBSDEBP for short), in which we replace the small jumps by a Brownian motion. Then, we prove the convergence of the approximation when the size of small jumps ε goes to 0. In the second step we obtain the Lp-Hölder continuity of the solution of the FBSDEBP and we construct two numerical schemes for this FBSDEBP. Based on the Lp-Hölder estimate, we prove the convergence of the scheme when the number of time steps n goes to ∞. Combining these two steps leads to the proof of the convergence of numerical schemes to the solution of FBSDEs driven by pure jump Lévy processes.

Keywords

Discrete-time approximationEuler schemedecoupled forward‒backward SDE with jumpssmall jumpsMalliavin calculus

MSC classification

Primary: 60H35: Computational methods for stochastic equations
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60J75: Jump processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 3 , September 2013 , pp. 791 - 821
Copyright
© Applied Probability Trust 

Footnotes

Supported by the Marie Curie Initial Training Network (ITN) project ‘Deterministic and Stochastic Controlled Systems and Application’ FP7-PEOPLE-2007-1-1-ITN, no. 213841-2.

References

Applebaum, D. (2004). Lévy Processes and Stochastic Calculus (Camb. Stud. Adv. Math. 93). Cambridge University Press.Google Scholar
Asmussen, S. and Rosiński, J. (2001). Approximations of small Jumps of Levy processes with a view towards simulation. J. Appl. Prob. 38, 482–493.Google Scholar
Bally, V. and Pagäs, G. (2003). Error analysis of the optimal quantization algorithm for obstacle problems. Stoch. Process. Appl. 106, 1–40.Google Scholar
Barles, G., Buckdahn, R. and Pardoux, E. (1997). Backward stochastic differential equations and integral-partial differential equations. Stoch. Stoch. Reports 60, 57–83.Google Scholar
Bouchard, B. and Elie, R. (2008). Discrete-time approximation of decoupled forward-backward SDE with Jumps. Stoch. Process. Appl. 118, 53–75.Google Scholar
Bouchard, B. and Touzi, N. (2004). Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations. Stoch. Process. Appl. 111, 175–206.Google Scholar
Briand, P., Delyon, B. and Mémin, J. (2001). Donsker-type theorem for BSDEs. Electron. Commun. Prob. 6, 1–14.CrossRefGoogle Scholar
Chevance, D. (1997). Numerical methods for backward stochastic differential equations. In Numerical Methods in Finance, eds Rogers, L. C. G. and Talay, D., Cambridge University Press, pp. 232–244.CrossRefGoogle Scholar
Clément, E., Lamberton, D. and Protter, P. (2002). An analysis of a least squares regression method for American option pricing. Finance Stoch. 6, 449–472.Google Scholar
Coquet, F., Mackevicius, V. and Mémin, J. (1998). Stability in D of martingales and backward equations under discretization of filtration. Stoch. Process. Appl. 75, 235–248.Google Scholar
Delong, Ł. and Imkeller, P. (2010). On Malliavin's differentiability of BSDEs with time delayed generators driven by Brownian motions and Poisson random measures. Stoch. Process. Appl. 120, 1748–1775.Google Scholar
Douglas, J. Jr., Ma, J. and Protter, P. (1996). Numerical methods for forward-backward stochastic differential equations. Ann. Appl. Prob. 6, 940–968.Google Scholar
El Karoui, N., Peng, S. and Quenez, M. C. (1997). Backward stochastic differential equations in finance. Math. Finance 7, 1–71.Google Scholar
Es-Sebaiy, K. and Tudor, C. A. (2008). Lévy processes and Itô Skorokhod integrals. Theory Stoch. Process. 14, 10–18.Google Scholar
Fujiwara, T. and Kunita, H. (1985). Stochastic differential equations of Jump type and Lévy processes in diffeomorphism group. J. Math. Kyoto Univ. 25, 71–106.Google Scholar
Hu, Y., Nualart, D. and Song, X. (2011). Malliavin calculus for backward stochastic differential equations and application to numerical solutions. Ann. Appl. Prob. 21, 2379–2424.Google Scholar
Kohatsu-Higa, A. and Tankov, P. (2010). Jump-adapted discretization schemes for Levy-driven SDEs. Stoch. Process. Appl. 120, 2258–2285.Google Scholar
Lemor, J. P., Gobet, E. and Warin, X. (2006). Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations. Bernoulli 12, 889–916.CrossRefGoogle Scholar
Ma, J., Protter, P. and Yong, J. M. (1994). Solving forward-backward stochastic differential equations—a four step scheme. Prob. Theory Relat. Fields 98, 339–359.Google Scholar
Nualart, D. (1995). The Malliavin Calculus and Related Topics. Springer, New York.CrossRefGoogle Scholar
Petrou, E. (2008). Malliavin calculus in Lévy spaces and applications to finance. Electron. J. Prob. 13, 852–879.Google Scholar
Solé, J. L., Utzet, F. and Vives, J. (2007). Canonical Lévy process and Malliavin calculus. Stoch. Process. Appl. 117, 165–187.Google Scholar