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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.

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