A Probabilistic numerical method for fully nonlinear parabolic pdes

Annals of Applied Probability

Volumn 21, Issue 4, 2011, Pages 1322-1364

  Fahim, Arash ^a   Touzi, Nizar ^a   Warin, Xavier ^b  

^a CENTRE DE MATHÉMATIQUES APPLIQUÉES   (France)

^b EDF R AND D   (France)

Author keywords

Monotone schemes; Monte Carlo approximation; Second order backward stochastic differential equations; Viscosity solutions

Indexed keywords

EID: 80051613006     PISSN: 10505164     EISSN: None     Source Type: Journal    
DOI: 10.1214/10-AAP723     Document Type: Article

References (30)

1. ANDERSEN, L. B. G. and BROTHERTON-RATCLIFFE, R. (2005). Extended LIBOR market models with stochastic üolatility. Journal of Computational Finance 9 1-40.
2. BALLY, V. (1997). Approximation scheme for solutions of BSDE. In Backward Stochastic Differential Equations (Paris, 1995-1996). Pitman Res. Notes Math. Ser. 364 177-191. Longman, Harlow. MR1752682
3. BARLES, G. and JAKOBSEN, E. R. (2002). On the conüergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations. M2AN Math. Model. Numer. Anal. 36 33-54. MR1916291
4. BARLES, G. and JAKOBSEN, E. R. (2005). Error bounds for monotone approximation schemes for Hamilton-Jacobi-Bellman equations. SIAM J. Numer. Anal. 43 540-558 (electronic). MR2177879
5. 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
6. BARLES, G. and SOUGANIDIS, P. E. (1991). Conüergence of approximation schemes for fully nonlinear second order equations. Asymptot. Anal. 4 271-283. MR1115933
7. 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 (electronic). MR2005192
8. BOUCHARD, B., EKELAND, I. and TOUZI, N. (2004). On the Malliaüin approach to Monte Carlo approximation of conditional expectations. Finance Stoch. 8 45-71. MR2022978 (Pubitemid 41239194)
9. 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 (Pubitemid 38561889)
10. CAMILLI, F. and FALCONE, M. (1995). An approximation scheme for the optimal control of diffusion processes. RAIRO Modél. Math. Anal. Numér. 29 97-122. MR1326802
11. CAMILLI, F. and JAKOBSEN, E. R. (2009). A finite element like scheme for integro-partial differential Hamilton-Jacobi-Bellman equations. SIAM J. Numer. Anal. 47 2407-2431. MR2525605
12. CHERIDITO, P., SONER, H. M., TOUZI, N. and VICTOIR, N. (2007). Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs. Comm. Pure Appl. Math. 60 1081-1110. MR2319056
13. CHEVANCE, D. (1996). Résolution numériques des équations différentielles stochastiques rétrogrades. Ph.D. thesis, Uniü. Proüence.
14. CRISAN, D., MANOLARAKIS, K. and TOUZI, N. (2010). On the Monte Carlo simulation of BSDEs: An improüement on the Malliaüin weights. Stochastic Process. Appl. 120 1133-1158. MR2639741
15. DEBRABANT, K. and JAKOBSEN, E. R. (2009). Semi-Lagrangian schemes for linear and fully non-linear diffusion equations. Preprint.
16. EL KAROUI, N., PENG, S. andQUENEZ, M. C. (1997). Backward stochastic differential equations in finance. Math. Finance 7 1-71. MR1434407 (Pubitemid 127342795)
17. GOBET, E., LEMOR, J.-P. and WARIN, X. (2005). A regression-based Monte Carlo method to solüe backward stochastic differential equations. Ann. Appl. Probab. 15 2172-2202. MR2152657 (Pubitemid 43205557)
18. HESTON, S. L. (1993). A closed-form solution for options with stochastic üolatility with applications to bond and currency options. The Reüiew of Financial Studies 6 327-343.
19. KAHL, C. and JÄCKEL, P. (2006). Fast strong approximation Monte Carlo schemes for stochastic üolatility models. Quant. Finance 6 513-536. MR2278638 (Pubitemid 44837712)
20. KOHN, R. V. and SERFATY, S. (2010). A deterministic-control-based approach to fully nonlinear parabolic and elliptic equations. Comm. Pure Appl. Math. 63 1298-1350. MR2681474
21. KRYLOV, N. V. (1997). On the rate of conüergence of finite-difference approximations for Bellman's equations. Algebra i Analiz 9 245-256. MR1466804
22. KRYLOV, N. V. (2000). On the rate of conüergence of finite-difference approximations for Bellman's equations with üariable coefficients. Probab. Theory Related Fields 117 1-16. MR1759507
23. KRYLOV, N. V. (2005). The rate of conüergence of finite-difference approximations for Bellman equations with Lipschitz coefficients. Appl. Math. Optim. 52 365-399. MR2174020 (Pubitemid 43022911)
24. LIONS, P. L. and REGNIER, H. (2001). Calcul du prix et des sensibilités d'une option américaine par une méthode de Monte Carlo. Preprint.
25. LONGSTAFF, F. A. and SCHWARTZ, R. S. (2001). Valuing American options by simulation: A simple least-square approach. The Reüiew of Financial Studies 14 113-147. (Pubitemid 33587386)
26. LORD, R., KOEKKOEK, R. and VAN DIJK, D. (2010). A comparison of biased simulation schemes for stochastic üolatility models. Quant. Finance 10 177-194. MR2642962
27. MUNOS, R. and ZIDANI, H. (2005). Consistency of a simple multidimensional scheme for Hamilton-Jacobi-Bellman equations. C. R. Math. Acad. Sci. Paris 340 499-502. MR2135230 (Pubitemid 40440670)
28. SONER, H. M. and TOUZI, N. (2003). A stochastic representation for mean curüature type geometric flows. Ann. Probab. 31 1145-1165. MR1988466 (Pubitemid 36881391)
29. ZARIPHOPOULOU, T. (2001). A solution approach to üaluation with unhedgeable risks. Finance Stoch. 5 61-82. MR1807876 (Pubitemid 33218405)
30. ZHANG, J. (2004). A numerical scheme for BSDEs. Ann. Appl. Probab. 14 459-488. MR2023027 (Pubitemid 41449263)