The Annals of Applied Probability

Discrete-time approximation of multidimensional BSDEs with oblique reflections

Jean-Francois Chassagneux, Romuald Elie, and Idris Kharroubi

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Abstract

In this paper, we study the discrete-time approximation of multidimensional reflected BSDEs of the type of those presented by Hu and Tang [Probab. Theory Related Fields 147 (2010) 89–121] and generalized by Hamadène and Zhang [Stochastic Process. Appl. 120 (2010) 403–426]. In comparison to the penalizing approach followed by Hamadène and Jeanblanc [Math. Oper. Res. 32 (2007) 182–192] or Elie and Kharroubi [Statist. Probab. Lett. 80 (2010) 1388–1396], we study a more natural scheme based on oblique projections. We provide a control on the error of the algorithm by introducing and studying the notion of multidimensional discretely reflected BSDE. In the particular case where the driver does not depend on the variable Z, the error on the grid points is of order ½ − ε, ε > 0.

Article information

Source
Ann. Appl. Probab., Volume 22, Number 3 (2012), 971-1007.

Dates
First available in Project Euclid: 18 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1337347536

Digital Object Identifier
doi:10.1214/11-AAP771

Mathematical Reviews number (MathSciNet)
MR2977983

Zentralblatt MATH identifier
1243.93128

Subjects
Primary: 93E20: Optimal stochastic control 65C99: None of the above, but in this section
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.)

Keywords
BSDE with oblique reflections discrete time approximation switching problems

Citation

Chassagneux, Jean-Francois; Elie, Romuald; Kharroubi, Idris. Discrete-time approximation of multidimensional BSDEs with oblique reflections. Ann. Appl. Probab. 22 (2012), no. 3, 971--1007. doi:10.1214/11-AAP771. https://projecteuclid.org/euclid.aoap/1337347536


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References

  • [1] Bouchard, B. and Chassagneux, J.-F. (2008). Discrete-time approximation for continuously and discretely reflected BSDEs. Stochastic Process. Appl. 118 2269–2293.
  • [2] Bouchard, B. and Elie, R. (2008). Discrete-time approximation of decoupled forward–backward SDE with jumps. Stochastic Process. Appl. 118 53–75.
  • [3] Bouchard, B., Elie, R. and Touzi, N. (2009). Discrete-time approximation of BSDEs and probabilistic schemes for fully nonlinear PDEs. In Advanced Financial Modelling. Radon Ser. Comput. Appl. Math. 8 91–124. de Gruyter, Berlin.
  • [4] Bouchard, B. and Touzi, N. (2004). Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations. Stochastic Process. Appl. 111 175–206.
  • [5] Carmona, R. and Ludkovski, M. (2008). Pricing asset scheduling flexibility using optimal switching. Appl. Math. Finance 15 405–447.
  • [6] Chassagneux, J. F. (2008). Processus réfléchis en finance et probabilité numérique. Ph.D. thesis, Univ. Paris 7.
  • [7] Chassagneux, J.-F. (2009). A discrete-time approximation for doubly reflected BSDEs. Adv. in Appl. Probab. 41 101–130.
  • [8] Chassagneux, J. F., Elie, R. and Kharroubi, I. (2011). A note on the existence and uniqueness of solutions of reflected BSDEs associated to switching problems. Electron. Commun. Probab. 16 120–128.
  • [9] Djehiche, B., Hamadène, S. and Popier, A. (2009). A finite horizon optimal multiple switching problem. SIAM J. Control Optim. 48 2751–2770.
  • [10] El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S. and Quenez, M. C. (1997). Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s. Ann. Probab. 25 702–737.
  • [11] Elie, R. and Kharroubi, I. (2010). Adding constraints to BSDEs with jumps: An alternative to multidimensional reflections. Preprint CEREMADE.
  • [12] Elie, R. and Kharroubi, I. (2010). Probabilistic representation and approximation for coupled systems of variational inequalities. Statist. Probab. Lett. 80 1388–1396.
  • [13] Hamadène, S. and Jeanblanc, M. (2007). On the starting and stopping problem: Application in reversible investments. Math. Oper. Res. 32 182–192.
  • [14] Hamadène, S. and Zhang, J. (2010). Switching problem and related system of reflected backward SDEs. Stochastic Process. Appl. 120 403–426.
  • [15] Hu, Y. and Tang, S. (2010). Multi-dimensional BSDE with oblique reflection and optimal switching. Probab. Theory Related Fields 147 89–121.
  • [16] Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Applications of Mathematics (New York) 23. Springer, Berlin.
  • [17] Kunita, H. (1990). Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics 24. Cambridge Univ. Press, Cambridge.
  • [18] Ma, J. and Zhang, J. (2005). Representations and regularities for solutions to BSDEs with reflections. Stochastic Process. Appl. 115 539–569.
  • [19] Nualart, D. (1995). The Malliavin Calculus and Related Topics. Springer, New York.
  • [20] Pardoux, É. and Peng, S. G. (1990). Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 55–61.
  • [21] Peng, S. (1999). Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob–Meyer’s type. Probab. Theory Related Fields 113 473–499.
  • [22] Pham, H., Ly Vath, V. and Zhou, X. Y. (2009). Optimal switching over multiple regimes. SIAM J. Control Optim. 48 2217–2253.
  • [23] Porchet, A., Touzi, N. and Warin, X. (2009). Valuation of a powerplant under production constraints and markets incompleteness. Math. Methods Oper. Res. 70 47–75.
  • [24] Zhang, J. (2004). A numerical scheme for BSDEs. Ann. Appl. Probab. 14 459–488.