Discrete time approximation of fully nonlinear HJB equations via BSDES with nonpositive jumps

Annals of Applied Probability

Volumn 25, Issue 4, 2015, Pages 2301-2338

  Kharroubi, Idris ^a   Langrené, Nicolas ^b   Pham, Huyên ^b  

^a UNIVERSITÉ PARIS DAUPHINE   (France)

^b UNIVERSITÉ PARIS DESCARTES   (France)

Author keywords

Backward stochastic differential equations; Discrete time approximation; Hamilton Jacobi Bellman equation; Nonlinear degenerate PDE; Optimal control; Sample

Indexed keywords

EID: 84936797237     PISSN: 10505164     EISSN: None     Source Type: Journal    
DOI: 10.1214/14-AAP1049     Document Type: Article

References (23)

1. BALLY, V. and PAGÈS, G. (2003). Error analysis of the optimal quantization algorithm for obstacle problems. Stochastic Process. Appl. 106 1-40. MR1983041
2. BARLES, G., BUCKDAHN, R. and PARDOUX, E. (1997). Backward stochastic differential equations and integral-partial differential equations. Stochastics Stochastics Rep. 60 57-83. MR1436432
3. BARLES, G. and JAKOBSEN, E. R. (2007). Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations. Math. Comp. 76 1861-1893 (electronic). MR2336272
4. BONNANS, J. F. and ZIDANI, H. (2003). Consistency of generalized finite difference schemes for the stochastic HJB equation. SIAM J. Numer. Anal. 41 1008-1021. MR2005192
5. BOUCHARD, B. and CHASSAGNEUX, J.-F. (2008). Discrete-time approximation for continuously and discretely reflected BSDEs. Stochastic Process. Appl. 118 2269-2293. MR2474351
6. BOUCHARD, B. and ELIE, R. (2008). Discrete-time approximation of decoupled forward-backward SDE with jumps. Stochastic Process. Appl. 118 53-75. MR2376252
7. BOUCHARD, B. and TOUZI, N. (2004). Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations. Stochastic Process. Appl. 111 175-206. MR2056536
8. CRANDALL, M. G., ISHII, H. and LIONS, P.-L. (1992). User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 1-67. MR1118699
9. FAHIM, A., TOUZI, N. and WARIN, X. (2011). A probabilistic numerical method for fully nonlinear parabolic PDEs. Ann. Appl. Probab. 21 1322-1364. MR2857450
10. FLEMING, W. H. and SONER, H. M. (2006). Controlled Markov Processes and Viscosity Solutions, 2nd ed. Stochastic Modelling and Applied Probability 25. Springer, New York. MR2179357
11. GUO, W., ZHANG, J. and ZHUO, J. (2006). A monotone scheme for high dimensional fully nonlinear PDE. Preprint.
12. KHARROUBI, I., LANGRENÉ, N. and PHAM, H. (2014). A numerical algorithm for fully non-linear HJB equations: An approach by control randomization. Monte Carlo Methods Appl. 20 145-165. MR3213592
13. KHARROUBI, I. and PHAM, H. (2014). Feynman-Kac representation for Hamilton-Jacobi-Bellman IPDE. Ann. Probab. To appear.
14. KLOEDEN, P. E. and PLATEN, E. (1992). Numerical Solution of Stochastic Differential Equations. Applications of Mathematics (New York) 23. Springer, Berlin. MR1214374
15. KRYLOV, N. V. (1999). Approximating value functions for controlled degenerate diffusion processes by using piece-wise constant policies. Electron. J. Probab. 4 1-19. MR1668597
16. KRYLOV, N. V. (2000). On the rate of convergence of finite-difference approximations for Bellman's equations with variable coefficients. Probab. Theory Related Fields 117 1-16. MR1759507
17. KUSHNER, H. J. and DUPUIS, P. G. (1992). Numerical Methods for Stochastic Control Problems in Continuous Time. Applications of Mathematics (New York) 24. Springer, New York. MR1217486
18. 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. MR2265667
19. PHAM, H. (2009). Continuous-Time Stochastic Control and Optimization with Financial Applications. Stochastic Modelling and Applied Probability 61. Springer, Berlin. MR2533355
20. POSSAMAÏ, D. and TAN, X. (2013). Weak approximation of second order BSDEs. Preprint.
21. SONER, H. M., TOUZI, N. and ZHANG, J. (2012). Wellposedness of second order backward SDEs. Probab. Theory Related Fields 153 149-190. MR2925572
22. TAN, X. (2013). A splitting method for fully nonlinear degenerate parabolic PDEs. Electron. J. Probab. 18 24. MR3035743
23. ZHANG, J. (2004). A numerical scheme for BSDEs. Ann. Appl. Probab. 14 459-488. MR2023027