Advances in Applied Probability

Discrete-time approximation of decoupled forward‒backward stochastic differential equations driven by pure jump Lévy processes

Soufiane Aazizi

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Abstract

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.

Article information

Source
Adv. in Appl. Probab., Volume 45, Number 3 (2013), 791-821.

Dates
First available in Project Euclid: 30 August 2013

Permanent link to this document
https://projecteuclid.org/euclid.aap/1377868539

Digital Object Identifier
doi:10.1239/aap/1377868539

Mathematical Reviews number (MathSciNet)
MR3102472

Zentralblatt MATH identifier
1274.60216

Subjects
Primary: 60H35: Computational methods for stochastic equations [See also 65C30]
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60J75: Jump processes

Keywords
Discrete-time approximation Euler scheme decoupled forward‒backward SDE with jumps mall jumps Malliavin calculus

Citation

Aazizi, Soufiane. Discrete-time approximation of decoupled forward‒backward stochastic differential equations driven by pure jump Lévy processes. Adv. in Appl. Probab. 45 (2013), no. 3, 791--821. doi:10.1239/aap/1377868539. https://projecteuclid.org/euclid.aap/1377868539


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