Stochastic representations for solutions to parabolic Dirichlet problems for nonlocal Bellman equations. (English) Zbl 1469.35212
Summary: We prove a stochastic representation formula for the viscosity solution of Dirichlet terminal-boundary value problem for a degenerate Hamilton-Jacobi-Bellman integro-partial differential equation in a bounded domain. We show that the unique viscosity solution is the value function of the associated stochastic optimal control problem. We also obtain the dynamic programming principle for the associated stochastic optimal control problem in a bounded domain.
MSC:
| 35R09 | Integro-partial differential equations |
| 35K61 | Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations |
| 35K65 | Degenerate parabolic equations |
| 49L20 | Dynamic programming in optimal control and differential games |
| 49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 60H30 | Applications of stochastic analysis (to PDEs, etc.) |
| 93E20 | Optimal stochastic control |
Keywords:
viscosity solution; integro-PDE; Hamilton-Jacobi-Bellman equation; dynamic programming principle; stochastic representation formula; value function; Lévy processReferences:
| [1] | Applebaum, D. (2009). Lévy Processes and Stochastic Calculus, 2nd ed. Cambridge Studies in Advanced Mathematics 116. Cambridge Univ. Press, Cambridge. · Zbl 1200.60001 |
| [2] | Arapostathis, A., Biswas, A. and Caffarelli, L. (2016). The Dirichlet problem for stable-like operators and related probabilistic representations. Comm. Partial Differential Equations 41 1472-1511. · Zbl 1352.60080 · doi:10.1080/03605302.2016.1207084 |
| [3] | Barles, G., Buckdahn, R. and Pardoux, E. (1997). Backward stochastic differential equations and integral-partial differential equations. Stoch. Stoch. Rep. 60 57-83. · Zbl 0878.60036 · doi:10.1080/17442509708834099 |
| [4] | Barles, G. and Imbert, C. (2008). Second-order elliptic integro-differential equations: Viscosity solutions’ theory revisited. Ann. Inst. H. Poincaré Anal. Non Linéaire 25 567-585. · Zbl 1155.45004 · doi:10.1016/j.anihpc.2007.02.007 |
| [5] | Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge. |
| [6] | Biswas, I. H. (2012). On zero-sum stochastic differential games with jump-diffusion driven state: A viscosity solution framework. SIAM J. Control Optim. 50 1823-1858. · Zbl 1253.91027 · doi:10.1137/080720504 |
| [7] | Biswas, I. H., Jakobsen, E. R. and Karlsen, K. H. (2010). Viscosity solutions for a system of integro-PDEs and connections to optimal switching and control of jump-diffusion processes. Appl. Math. Optim. 62 47-80. · Zbl 1197.49028 · doi:10.1007/s00245-009-9095-8 |
| [8] | Bounkhel, M. (2012). Regularity Concepts in Nonsmooth Analysis: Theory and Applications. Springer Optimization and Its Applications 59. Springer, New York. · Zbl 1259.49022 |
| [9] | Buckdahn, R., Hu, Y. and Li, J. (2011). Stochastic representation for solutions of Isaacs’ type integral-partial differential equations. Stochastic Process. Appl. 121 2715-2750. · Zbl 1243.91011 · doi:10.1016/j.spa.2011.07.011 |
| [10] | Buckdahn, R. and Li, J. (2008). Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations. SIAM J. Control Optim. 47 444-475. · Zbl 1157.93040 · doi:10.1137/060671954 |
| [11] | Clarke, F. H., Ledyaev, Yu. S., Stern, R. J. and Wolenski, P. R. (1998). Nonsmooth Analysis and Control Theory. Graduate Texts in Mathematics 178. Springer, New York. · Zbl 1047.49500 |
| [12] | Clarke, F. H., Stern, R. J. and Wolenski, P. R. (1995). Proximal smoothness and the lower-\(C^2\) property. J. Convex Anal. 2 117-144. · Zbl 0881.49008 |
| [13] | El Karoui, N., H Nguyen, D. and Jeanblanc-Picqué, M. (1987). Compactification methods in the control of degenerate diffusions: Existence of an optimal control. Stochastics 20 169-219. · Zbl 0613.60051 · doi:10.1080/17442508708833443 |
| [14] | Feynman, R. P. (1948). Space-time approach to non-relativistic quantum mechanics. Rev. Modern Phys. 20 367-387. · Zbl 1371.81126 · doi:10.1103/RevModPhys.20.367 |
| [15] | 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. · Zbl 1105.60005 |
| [16] | Fleming, W. H. and Souganidis, P. E. (1989). On the existence of value functions of two-player, zero-sum stochastic differential games. Indiana Univ. Math. J. 38 293-314. · Zbl 0686.90049 · doi:10.1512/iumj.1989.38.38015 |
| [17] | Garroni, M. G. and Menaldi, J. L. (2002). Second Order Elliptic Integro-Differential Problems. Chapman & Hall/CRC Research Notes in Mathematics 430. CRC Press/CRC, Boca Raton, FL. · Zbl 1014.45002 |
| [18] | Glau, K. (2016). A Feynman-Kac-type formula for Lévy processes with discontinuous killing rates. Finance Stoch. 20 1021-1059. · Zbl 1355.60060 · doi:10.1007/s00780-016-0301-7 |
| [19] | Ishikawa, Y. (2004). Optimal control problem associated with jump processes. Appl. Math. Optim. 50 21-65. · Zbl 1052.60050 · doi:10.1007/s00245-004-0795-9 |
| [20] | Jakobsen, E. R. and Karlsen, K. H. (2006). A “maximum principle for semicontinuous functions” applicable to integro-partial differential equations. NoDEA Nonlinear Differential Equations Appl. 13 137-165. · Zbl 1105.45006 · doi:10.1007/s00030-005-0031-6 |
| [21] | Kac, M. (1949). On distributions of certain Wiener functionals. Trans. Amer. Math. Soc. 65 1-13. · Zbl 0032.03501 · doi:10.1090/S0002-9947-1949-0027960-X |
| [22] | Katsoulakis, M. A. (1995). A representation formula and regularizing properties for viscosity solutions of second-order fully nonlinear degenerate parabolic equations. Nonlinear Anal. 24 147-158. · Zbl 0824.35057 · doi:10.1016/0362-546X(94)00170-M |
| [23] | Kharroubi, I. and Pham, H. (2015). Feynman-Kac representation for Hamilton-Jacobi-Bellman IPDE. Ann. Probab. 43 1823-1865. · Zbl 1333.60150 |
| [24] | Koike, S. and Święch, A. (2013). Representation formulas for solutions of Isaacs integro-PDE. Indiana Univ. Math. J. 62 1473-1502. · Zbl 1292.49030 · doi:10.1512/iumj.2013.62.5109 |
| [25] | Kovats, J. (2009). Value functions and the Dirichlet problem for Isaacs equation in a smooth domain. Trans. Amer. Math. Soc. 361 4045-4076. · Zbl 1186.49021 · doi:10.1090/S0002-9947-09-04732-1 |
| [26] | Krylov, N. V. (2009). Controlled Diffusion Processes. Stochastic Modelling and Applied Probability 14. Springer, Berlin. |
| [27] | Lions, P.-L. (1983). Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. I. The dynamic programming principle and applications. Comm. Partial Differential Equations 8 1101-1174. · Zbl 0716.49022 · doi:10.1080/03605308308820297 |
| [28] | Lions, P.-L. (1983). Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. III. Regularity of the optimal cost function. In Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, Vol. V (Paris, 1981/1982). Res. Notes in Math. 93 95-205. Pitman, Boston, MA. · Zbl 0716.49024 |
| [29] | Ma, J. and Zhang, J. (2002). Representation theorems for backward stochastic differential equations. Ann. Appl. Probab. 12 1390-1418. · Zbl 1017.60067 · doi:10.1214/aoap/1037125868 |
| [30] | Mou, C. (2019). Remarks on Schauder estimates and existence of classical solutions for a class of uniformly parabolic Hamilton-Jacobi-Bellman integro-PDEs. J. Dynam. Differential Equations. 31 719-743. · Zbl 1414.35253 · doi:10.1007/s10884-018-9649-z |
| [31] | Mou, C. (2017). Perron’s method for nonlocal fully nonlinear equations. Anal. PDE 10 1227-1254. · Zbl 1370.35083 · doi:10.2140/apde.2017.10.1227 |
| [32] | Mou, C. (2018). Existence of \(C^{\alpha }\) solutions to integro-PDEs. Preprint. · Zbl 1432.35215 · doi:10.1007/s00526-019-1597-x |
| [33] | Nisio, M. (2015). Stochastic Control Theory: Dynamic Programming Principle, 2nd ed. Probability Theory and Stochastic Modelling 72. Springer, Tokyo. · Zbl 1306.93077 |
| [34] | Nualart, D. and Schoutens, W. (2001). Backward stochastic differential equations and Feynman-Kac formula for Lévy processes, with applications in finance. Bernoulli 7 761-776. · Zbl 0991.60045 · doi:10.2307/3318541 |
| [35] | Øksendal, B. and Sulem, A. (2007). Applied Stochastic Control of Jump Diffusions, 2nd ed. Universitext. Springer, Berlin. · Zbl 1116.93004 |
| [36] | Pardoux, É. and Peng, S. (1992). Backward stochastic differential equations and quasilinear parabolic partial differential equations. In Stochastic Partial Differential Equations and Their Applications (Charlotte, NC, 1991). Lect. Notes Control Inf. Sci. 176 200-217. Springer, Berlin. · Zbl 0766.60079 |
| [37] | Peszat, S. and Zabczyk, J. (2007). Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach. Encyclopedia of Mathematics and Its Applications 113. Cambridge Univ. Press, Cambridge. · Zbl 1205.60122 |
| [38] | Pham, H. (1998). Optimal stopping of controlled jump diffusion processes: A viscosity solution approach. J. Math. Systems Estim. Control 8 27 pp. · Zbl 0899.60039 |
| [39] | Poliquin, R. A., Rockafellar, R. T. and Thibault, L. (2000). Local differentiability of distance functions. Trans. Amer. Math. Soc. 352 5231-5249. · Zbl 0960.49018 · doi:10.1090/S0002-9947-00-02550-2 |
| [40] | Seydel, R. C. (2009). Existence and uniqueness of viscosity solutions for QVI associated with impulse control of jump-diffusions. Stochastic Process. Appl. 119 3719-3748. · Zbl 1180.35164 · doi:10.1016/j.spa.2009.07.004 |
| [41] | Soner, H. M. (1986). Optimal control with state-space constraint. II. SIAM J. Control Optim. 24 1110-1122. · Zbl 0619.49013 · doi:10.1137/0324067 |
| [42] | Soner, H. M. (1988). Optimal control of jump-Markov processes and viscosity solutions. In Stochastic Differential Systems, Stochastic Control Theory and Applications (Minneapolis, Minn., 1986). IMA Vol. Math. Appl. 10 501-511. Springer, New York. · Zbl 0850.93889 |
| [43] | Święch, A. (1996). Another approach to the existence of value functions of stochastic differential games. J. Math. Anal. Appl. 204 884-897. · Zbl 0870.90107 |
| [44] | Święch, A. and Zabczyk, J. (2016). Integro-PDE in Hilbert spaces: Existence of viscosity solutions. Potential Anal. 45 703-736. · Zbl 1352.49025 · doi:10.1007/s11118-016-9563-0 |
| [45] | Touzi, N. (2004). Stochastic Control Problems, Viscosity Solutions and Application to Finance. Scuola Normale Superiore di Pisa. Quaderni. [Publications of the Scuola Normale Superiore of Pisa]. Scuola Normale Superiore, Pisa. · Zbl 1076.93001 |
| [46] | Touzi, N. (2013). Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE. With Chapter 13 by Angès Tourin. Fields Institute Monographs 29. Springer, New York. · Zbl 1256.93008 |
| [47] | Yong, J. and Zhou, X. Y. (1999). Stochastic Controls: Hamiltonian Systems and HJB Equations. Applications of Mathematics (New York) 43. Springer, New York. · Zbl 0943.93002 |
| [48] | Zhang, J. (2005). Representation of solutions to BSDEs associated with a degenerate FSDE. Ann. Appl. Probab. 15 1798-1831. · Zbl 1083.60051 · doi:10.1214/105051605000000232 |
| [49] | Zhu, C., Yin, G. and Baran, N. A. (2015). Feynman-Kac formulas for regime-switching jump diffusions and their applications. Stochastics 87 1000-1032. · Zbl 1337.60200 · doi:10.1080/17442508.2015.1019884 |
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