On error bounds for monotone approximation schemes for multi-dimensional Isaacs equations

Asymptotic Analysis

Volumn 49, Issue 3-4, 2006, Pages 249-273

  Jakobsen, Espen R ^a  

^a NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Norway)

Author keywords

Control scheme; Convergence rate; Finite difference method; Hamilton Jacobi Bellman Isaacs equation; Nonlinear degenerate elliptic equation; Obstacle problem; Penalization method; Variational inequality; Viscosity solution

Indexed keywords

APPROXIMATION THEORY; CONVERGENCE OF NUMERICAL METHODS; FINITE DIFFERENCE METHOD; NONLINEAR EQUATIONS; PROBLEM SOLVING; VISCOSITY;

CONTROL SCHEME; CONVERGENCE RATE; HAMILTON-JACOBI-BELLMAN-ISAACS EQUATION; NONLINEAR DEGENERATE ELLIPTIC EQUATION; OBSTACLE PROBLEM; PENALIZATION METHOD; VARIATIONAL INEQUALITY; VISCOSITY SOLUTION;

ERROR ANALYSIS;

EID: 33748920037     PISSN: 09217134     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (25)

1. A.L. Amadori, The obstacle problem for nonlinear integro-differential operators arising in option pricing, Quaderno IAC Q21-000, 2000.
2. G. Barles and E.R. Jakobsen, On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations, M2AN Math. Model. Numer. Anal. 36(1) (2002), 33-54.
3. G. Barles and E.R. Jakobsen, Error bounds for monotone approximation schemes for Hamilton-Jacobi-Bellman equations, SIAM J. Numer. Anal. 43(2) (2005), 540-558.
4. G. Barles and P.E. Souganidis, Convergence of approximation schemes for fully nonlinear second-order equations, Asymptotic Anal. 4(3) (1991), 271-283.
5. A. Bensoussan and J.-L. Lions, Applications of Variational Inequalities in Stochastic Control, North-Holland Publishing Co., Amsterdam, 1982.
6. F.E. Benth, K.H. Karlsen and K. Reikvam, A semilinear Black and Scholes partial differential equation for valuing American options, Finance Stock. 7(3) (2003), 277-298.
7. J.F. Bonnans, S. Maroso and H. Zidani, Error estimates for stochastic differential games: the adverse stopping case, IMA J. Numer. Anal. 26 (2006), 188-212.
8. F. Camilli and M. Falcone, An approximation scheme for the optimal control of diffusion processes, RAIRO Model. Math. Anal. Numér. 29(1), 97-122.
9. M.G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second-order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27(1) (1992), 1-67.
10. M.G. Crandall and P.-L. Lions, Two approximations of solutions of Hamilton-Jacobi equations, Math. Comp. 43(167) (1984), 1-19.
11. W.H. Fleming and H.M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, New York, 1993.
12. W.H. Fleming and P.E. Souganidis, On the existence of value functions of two-player, zero-sum stochastic differential games, Indiana Univ. Math. J. 38(2) (1989), 293-314.
13. H. Ishii, On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions, Funkcial. Ekvac. 38(1) (1995), 101-120.
14. H. Ishii, On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs, Comm. Pure Appl. Math. 42(1) (1989), 15-45.
15. 1, BIT 44(2) (2004), 269-285.
16. E.R. Jakobsen, On the rate of convergence of approximation schemes for Bellman equations associated with optimal stopping time problems, Math. Models Methods Appl. Sci. (M3AS) 13(5) (2003), 613-644.
17. E.R. Jakobsen and K.H. Karlsen, Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate parabolic equations, J. Differential Equations 183 (2002), 497-525.
18. E.R. Jakobsen and K.H. Karlsen, Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate elliptic equations, Electron. J. Differential Equations 2002(39) (2002), 1-10.
19. D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Reprint of the 1980 original, SIAM, Philadelphia, 2000.
20. N.V. Krylov, On the rate of convergence of finite-difference approximations for Bellman's equations, St. Petersburg Math. J. 9(3) (1997), 639-650.
21. N.V. Krylov, On the rate of convergence of finite-difference approximations for Bellman's equations with variable coefficients, Probab. Theory Ralated Fields 117 (2000), 1-16.
22. H.J. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, Springer, New York, 2001.
23. J.L. Menaldi, Some estimates for finite difference approximations, SIAM J. Control Optim. 27(3) (1989), 579-607.
24. H. Pham, Optimal stopping of controlled jump diffusion processes: a viscosity solution approach, J. Math. Systems Estim. Control 8(1) (1998).
25. P. Wilmott, S. Howison and J. Dewynne, The Mathematics of Financial Derivatives. A Student Introduction, Cambridge University Press, Cambridge, 1995.