Continuous dependence estimates for viscosity solutions of integro-PDEs

Journal of Differential Equations

Volumn 212, Issue 2, 2005, Pages 278-318

  Jakobsen, Espen R ^a   Karlsen, Kenneth H ^b,c  

^a NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Norway)

^b UNIVERSITY OF BERGEN   (Norway)

^c UNIVERSITY OF OSLO   (Norway)

Author keywords

Bellman equation; Continuous dependence estimate; Convergence rate; Isaacs equation; Nonlinear degenerate parabolic integro partial differential equation; Regularity; Vanishing viscosity method; Viscosity solution

Indexed keywords

EID: 16844376284     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jde.2004.06.021     Document Type: Article

References (50)

1. O. Alvarez A. Tourin Viscosity solutions of nonlinear integro-differential equations Ann. Inst. H. Poincaré Anal. Non Linéaire 13 3 1996 293-317
2. A.L. Amadori, The obstacle problem for nonlinear integro-differential operators arising in option pricing, Quaderno IAC Q21-000, 2000
3. A.L. Amadori, Nonlinear integro-differential evolution problems arising in option pricing: A viscosity solutions approach, J. Differential Integral Equations 16 (7) (2003) 787-811.
4. M. Bardi I. Capuzzo-Dolcetta Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations 1997 Birkhäuser Boston
5. G. Barles Solutions de viscosité des équations de Hamilton-Jacobi 1994 Springer Paris
6. G. Barles R. Buckdahn E. Pardoux Backward stochastic differential equations and integral-partial differential equations Stochastics Stochastics Rep. 60 1-2 1997 57-83
7. G. Barles 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
8. G. Barles, E.R. Jakobsen, Error bounds for monotone approximation schemes for Hamilton-Jacobi-Bellman equations, preprint.
9. O.E. Barndorff-Nielsen Exponentially decreasing distributions for the logarithm of particle size J. Roy. Statist. Soc. Ser. B 353 1977 401-419
10. O.E. Barndorff-Nielsen Processes of normal inverse Gaussian type Finance Stochastics 2 1 1998 41-68
11. F.E. Benth K.H. Karlsen K. Reikvam Optimal portfolio management rules in a non-Gaussian market with durability and intertemporal substitution Finance Stochastics 5 4 2001 447-467
12. F.E. Benth K.H. Karlsen K. Reikvam Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint a viscosity solution approach Finance Stochastics 5 3 2001 275-303
13. F.E. Benth K.H. Karlsen K. Reikvam Portfolio optimization in a Lévy market with intertemporal substitution and transaction costs Stochastics Stochastics Rep. 74 3-4 2002 517-569
14. 1,β-regularity of viscosity solutions via a continuous dependence result, 2002, preprint.
15. T. Chan Pricing contingent claims on stocks driven by Lévy processes Ann. Appl. Probab. 9 2 1999 504-528
16. B. Cockburn G. Gripenberg J.-O. Londen Continuous dependence on the nonlinearity of viscosity solutions of parabolic equations J. Differential Equations 170 1 2001 180-187
17. M.G. Crandall H. Ishii The maximum principle for semicontinuous functions Differential Integral Equations 3 6 1990 1001-1014
18. M.G. Crandall H. Ishii 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
19. M.G. Crandall P.-L. Lions Two approximations of solutions of Hamilton-Jacobi equations Math. Comp. 43 167 1984 1-19
20. M.G. Crandall P.-L. Lions Viscosity solutions of Hamilton-Jacobi equations Trans. Amer. Math. Soc. 277 1 1983 1-42
21. C.M. Dafermos Hyperbolic conservation laws in continuum physics, Grundlehren der Mathematischen Wissenschaften vol. 325 2000 Springer Berlin
22. E. Eberlein U. Keller Hyperbolic distributions in finance Bernoulli 1 1995 281-299
23. S. Elganjoui, Diploma Thesis, Department of Mathematics, University of Bergen, Norway, January 2001.
24. S. Elganjoui, K.H. Karlsen, A Markov chain approximation scheme for a singular investment-consumption problem with Lévy driven stock prices, Unpublished manuscript available at URL http://www.mi.uib.no/~kennethk/, 2002.
25. W.H. Fleming H.M. Soner Controlled Markov Processes and Viscosity Solutions 1993 Springer New York
26. W.H. Fleming 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
27. N.C. Framstad B. Øksendal A. Sulem Optimal consumption and portfolio in a jump diffusion market with proportional transaction costs J. Math. Econom. 35 2 2001 233-257
28. M.G. Garroni J.-L. Menaldi Green functions for second order parabolic integro-differential problems Pitman Research Notes in Mathematics Series, vol. 275 1992 Longman Scientific & Technical Harlow
29. M.G. Garroni J.L. Menaldi Second Order Elliptic Integro-differential Problems 2002 Chapman & Hall/CRC Boca Raton
30. G. Gripenberg Estimates for viscosity solutions of parabolic equations with Dirichlet boundary conditions Proc. Amer. Math. Soc. 130 12 2002 3651-3660
31. 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
32. 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
33. E.R. Jakobsen K.H. Karlsen Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate parabolic equations J. Differential Equations 183 2002 497-525
34. E.R. Jakobsen K.H. Karlsen Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate elliptic equations Electron. J. Differential Equations 2002 39 2002 1-10
35. E.R. Jakobsen, K.H. Karlsen, A maximum principle for semicontinuous functions applicable to integro-partial differential equations, 2003, preprint.
36. S. Koike On the rate of convergence of solutions in singular perturbation problems J. Math. Anal. Appl. 157 1 1991 243-253
37. N.V. Krylov Controlled Diffusion Processes 1980 Springer New York
38. N.V. Krylov On the rate of convergence of finite-difference approximations for Bellman's equations with variable coefficients Probab. Theory Rel. Fields 117 2000 1-16
39. J.-L. Lions Perturbations singuliéres dans les problèmes aux limites et en contrôle optimal Lecture Notes in Mathematics vol. 323 1973 Springer Berlin-New York
40. P.-L. Lions Generalized solutions of Hamilton-Jacobi equations Research Notes in Mathematics vol. 69 1982 Pitman Boston-London
41. R. Mikulyavichyus G. Pragarauskas Nonlinear potentials of the Cauchy-Dirichlet problem for the Bellman integro-differential equation Liet. Mat. Rink. 36 2 1996 178-218
42. H. Pham Optimal stopping, free boundary, and American option in a jump-diffusion model Appl. Math. Optim. 35 2 1997 145-164
43. H. Pham Optimal stopping of controlled jump diffusion processes a viscosity solution approach J. Math. Systems Estim. Control 8 1 1998
44. T.H. Rydberg The normal inverse Gaussian Lévy process simulation and approximation. Heavy tails and highly volatile phenomena Comm. Statist. Stochastic Models 13 4 1997 887-910
45. A. Sayah Équations d'Hamilton-Jacobi du premier ordre avec termes intégro-différentiels. I. Unicité des solutions de viscosité. I Comm. Partial Differential Equations 16 6-7 1991 1057-1074
46. A. Sayah Équations d'Hamilton-Jacobi du premier ordre avec termes intégro-différentiels. II. Existence de solutions de viscosité Comm. Partial Differential Equations 16 6-7 1991 1075-1093
47. A. Shiryaev Essentials of stochastic finance Advanced Series on Statistical Science and Applied Probability vol. 3 1999 World Scientific Publishing Co. Inc. River Edge, NJ
48. H.M. Soner Optimal control with state-space constraint. II SIAM J. Control Optim. 24 6 1986 1110-1122
49. H.M. Soner Optimal control of jump-Markov processes and viscosity solutions Stochastic Differential Systems Stochastic Control Theory and Applications (Minneapolis, Minn., 1986), IMA Mathematics and Applications, vol. 10 1988 Springer New York 501-511
50. P.E. Souganidis Existence of viscosity solutions of Hamilton-Jacobi equations J. Differential Equations 56 3 1985 345-390