The Annals of Probability

Feynman–Kac representation for Hamilton–Jacobi–Bellman IPDE

Idris Kharroubi and Huyên Pham

Full-text: Open access


We aim to provide a Feynman–Kac type representation for Hamilton–Jacobi–Bellman equation, in terms of forward backward stochastic differential equation (FBSDE) with a simulatable forward process. For this purpose, we introduce a class of BSDE where the jumps component of the solution is subject to a partial nonpositive constraint. Existence and approximation of a unique minimal solution is proved by a penalization method under mild assumptions. We then show how minimal solution to this BSDE class provides a new probabilistic representation for nonlinear integro-partial differential equations (IPDEs) of Hamilton–Jacobi–Bellman (HJB) type, when considering a regime switching forward SDE in a Markovian framework, and importantly we do not make any ellipticity condition. Moreover, we state a dual formula of this BSDE minimal solution involving equivalent change of probability measures. This gives in particular an original representation for value functions of stochastic control problems including controlled diffusion coefficient.

Article information

Ann. Probab., Volume 43, Number 4 (2015), 1823-1865.

Received: December 2012
Revised: November 2013
First available in Project Euclid: 3 June 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H30: Applications of stochastic analysis (to PDE, etc.) 35K55: Nonlinear parabolic equations 93E20: Optimal stochastic control

BSDE with jumps constrained BSDE regime-switching jump-diffusion Hamilton–Jacobi–Bellman equation nonlinear Integral PDE viscosity solutions inf-convolution semiconcave approximation


Kharroubi, Idris; Pham, Huyên. Feynman–Kac representation for Hamilton–Jacobi–Bellman IPDE. Ann. Probab. 43 (2015), no. 4, 1823--1865. doi:10.1214/14-AOP920.

Export citation


  • [1] Barles, G. (1994). Solutions de Viscosité des équations de Hamilton-Jacobi. Mathématiques & Applications (Berlin) [Mathematics & Applications] 17. Springer, Paris.
  • [2] Barles, G., Buckdahn, R. and Pardoux, E. (1997). Backward stochastic differential equations and integral-partial differential equations. Stochastics Stochastics Rep. 60 57–83.
  • [3] 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.
  • [4] Bouchard, B. and Touzi, N. (2004). Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations. Stochastic Process. Appl. 111 175–206.
  • [5] Cannarsa, P. and Sinestrari, C. (2004). Semiconcave Functions, Hamilton–Jacobi Equations, and Optimal Control. Progress in Nonlinear Differential Equations and Their Applications 58. Birkhäuser, Boston, MA.
  • [6] Choukroun, S., Cosso, A. and Pham, H. (2013). Reflected BSDEs with nonpositive jumps, and controller-and-stopper games. Preprint. Available at arXiv:1308.5511.
  • [7] Ekren, I., Touzi, N. and Zhang, J. (2013). Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part I. Preprint.
  • [8] El Karoui, N. (1981). Les aspects probabilistes du contrôle stochastique. Lect. Notes in Mathematics 876. Springer, Berlin.
  • [9] El Karoui, N., Peng, S. and Quenez, M. C. (1997). Backward stochastic differential equations in finance. Math. Finance 7 1–71.
  • [10] Essaky, E. H. (2008). Reflected backward stochastic differential equation with jumps and RCLL obstacle. Bull. Sci. Math. 132 690–710.
  • [11] Friedman, A. (1975). Stochastic Differential Equations and Applications, Vol. 1. Probability and Mathematical Statistics 28. Academic Press, New York.
  • [12] Fuhrman, M. and Pham, H. (2013). Dual and backward SDE representation for optimal control of non-Markovian SDEs. Preprint. Available at arXiv:1310.6943.
  • [13] Henry-Labordère, P. (2012). Counterparty risk valuation: A marked branching diffusion approach. Preprint. Available at arxiv:1203.2369v1.
  • [14] Kharroubi, I., Langrené, N. and Pham, H. (2013). Discrete time approximation of fully nonlinear HJB equations via BSDEs with nonpositive jumps. Preprint. Available at arXiv:1311.4505.
  • [15] Kharroubi, I., Langrené, N. and Pham, H. (2013). Numerical algorithm for fully nonlinear HJB equations: An approach by control randomization. Preprint. Available at arXiv:1311.4503.
  • [16] Kharroubi, I., Ma, J., Pham, H. and Zhang, J. (2010). Backward SDEs with constrained jumps and quasi-variational inequalities. Ann. Probab. 38 794–840.
  • [17] Øksendal, B. and Sulem, A. (2007). Applied Stochastic Control of Jump Diffusions, 2nd ed. Springer, Berlin.
  • [18] 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) (B. Rozovskiin and R. Sowers, eds.). Lecture Notes in Control and Inform. Sci. 176 200–217. Springer, Berlin.
  • [19] Pardoux, É. and Peng, S. G. (1990). Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 55–61.
  • [20] Peng, S. (2000). Monotonic limit theorem for BSDEs and non-linear Doob–Meyer decomposition. Probab. Theory Related Fields 16 225–234.
  • [21] Peng, S. (2007). $G$-expectation, $G$-Brownian motion and related stochastic calculus of Itô type. In Stochastic Analysis and Applications. Abel Symp. 2 541–567. Springer, Berlin.
  • [22] Pham, H. (1998). Optimal stopping of controlled jump diffusion processes: A viscosity solution approach. J. Math. Systems Estim. Control 8 27 pp. (electronic).
  • [23] Pham, H. (2009). Continuous-time Stochastic Control and Optimization with Financial Applications. Stochastic Modelling and Applied Probability 61. Springer, Berlin.
  • [24] Protter, P. and Shimbo, K. (2008). No arbitrage and general semimartingales. In Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz. Inst. Math. Stat. Collect. 4 267–283. IMS, Beachwood, OH.
  • [25] Royer, M. (2006). Backward stochastic differential equations with jumps and related non-linear expectations. Stochastic Process. Appl. 116 1358–1376.
  • [26] Soner, H. M., Touzi, N. and Zhang, J. (2012). Wellposedness of second order backward SDEs. Probab. Theory Related Fields 153 149–190.
  • [27] Tang, S. J. and Li, X. J. (1994). Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim. 32 1447–1475.