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**Probabilistic interpretation of HJB equations by the representation theorem for generators of BSDEs.**
*(English)*
Zbl 1461.60043

Summary: The purpose of this note is to propose a new approach for the probabilistic interpretation of Hamilton-Jacobi-Bellman equations associated with stochastic recursive optimal control problems, utilizing the representation theorem for generators of backward stochastic differential equations. The key idea of our approach for proving this interpretation lies in the identity between solutions and generators given by the representation theorem. Compared with existing methods, our approach seems to be a feasible unified method for different frameworks and be more applicable to general settings. This can also be regarded as a new application of such representation theorem.

### MSC:

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

35K20 | Initial-boundary value problems for second-order parabolic equations |

49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |

### Keywords:

backward stochastic differential equation; recursive optimal control problem; Hamilton-Jacobi-Bellman equation; representation theorem for generator
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\textit{L. Xiao} et al., Electron. Commun. Probab. 25, Paper No. 30, 10 p. (2020; Zbl 1461.60043)

### References:

[1] | Guy Barles, Rainer Buckdahn, and Etienne Pardoux, Backward stochastic differential equations and integral-partial differential equations, Stochastics Stochastics Rep. 60 (1997), no. 1-2, 57-83. · Zbl 0878.60036 |

[2] | Philippe Briand, François Coquet, Ying Hu, Jean Mémin, and Shige Peng, A converse comparison theorem for BSDEs and related properties of \(g\)-expectation, Electron. Comm. Probab. 5 (2000), no. 13, 101-117. · Zbl 0966.60054 |

[3] | Rainer Buckdahn and Ying Hu, Probabilistic interpretation of a coupled system of Hamilton-Jacobi-Bellman equations, J. Evol. Equ. 10 (2010), no. 3, 529-549. · Zbl 1239.35036 |

[4] | Rainer Buckdahn and Juan Li, Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations, SIAM J. Control Optim. 47 (2008), no. 1, 444-475. · Zbl 1157.93040 |

[5] | Rainer Buckdahn and Tianyang Nie, Generalized Hamilton-Jacobi-Bellman equations with Dirichlet boundary and stochastic exit time optimal control problem, SIAM J. Control Optim. 54 (2016), no. 2, 602-631. · Zbl 1345.49035 |

[6] | Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1-67. · Zbl 0755.35015 |

[7] | Nicole El Karoui, Shige Peng, and Marie Claire Quenez, Backward stochastic differential equations in finance, Math. Finance 7 (1997), no. 1, 1-71. · Zbl 0884.90035 |

[8] | Shengjun Fan, Long Jiang, and Yingying Xu, Representation theorem for generators of BSDEs with monotonic and polynomial-growth generators in the space of processes, Electron. J. Probab. 16 (2011), no. 27, 830-844. · Zbl 1225.60094 |

[9] | Long Jiang, Representation theorems for generators of backward stochastic differential equations and their applications, Stochastic Process. Appl. 115 (2005), no. 12, 1883-1903. · Zbl 1078.60043 |

[10] | Long Jiang, Convexity, translation invariance and subadditivity for \(g\)-expectations and related risk measures, Ann. Appl. Probab. 18 (2008), no. 1, 245-258. · Zbl 1145.60032 |

[11] | Juan Li and Shanjian Tang, Optimal stochastic control with recursive cost functionals of stochastic differential systems reflected in a domain, ESAIM Control Optim. Calc. Var. 21 (2015), no. 4, 1150-1177. · Zbl 1341.49020 |

[12] | Juan Li and Qingmeng Wei, Optimal control problems of fully coupled FBSDEs and viscosity solutions of Hamilton-Jacobi-Bellman equations, SIAM J. Control Optim. 52 (2014), no. 3, 1622-1662. · Zbl 1295.93076 |

[13] | Etienne Pardoux, BSDEs, weak convergence and homogenization of semilinear PDEs, Nonlinear analysis, differential equations and control (F. H. Clarke, R. J. Stern, and G. Sabidussi, eds.), pp. 503-549, NATO Sci. Ser. C Math. Phys. Sci., vol. 528, Kluwer Acad. Publ., Dordrecht, 1999. · Zbl 0959.60049 |

[14] | Etienne Pardoux and Shige Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14 (1990), no. 1, 55-61. · Zbl 0692.93064 |

[15] | Etienne Pardoux, Frédéric Pradeilles, and Zusheng Rao, Probabilistic interpretation of a system of semi-linear parabolic partial differential equations, Ann. Inst. H. Poincaré Probab. Statist. 33 (1997), no. 4, 467-490. · Zbl 0891.60054 |

[16] | Etienne Pardoux and Shanjian Tang, Forward-backward stochastic differential equations and quasilinear parabolic PDEs, Probab. Theory Related Fields 114 (1999), no. 2, 123-150. · Zbl 0943.60057 |

[17] | Etienne Pardoux and Shuguang Zhang, Generalized BSDEs and nonlinear Neumann boundary value problems, Probab. Theory Related Fields 110 (1998), no. 4, 535-558. · Zbl 0909.60046 |

[18] | Shige Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics Stochastics Rep. 37 (1991), no. 1-2, 61-74. · Zbl 0739.60060 |

[19] | Shige Peng, A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation, Stochastics Stochastics Rep. 38 (1992), no. 2, 119-134. · Zbl 0756.49015 |

[20] | Shige Peng, Backward stochastic differential equations — stochastic optimization theory and viscosity solutions of HJB equations, Topics on stochastic analysis (In Chinese) (Jiaan Yan, Shige Peng, Shizan Fang, and Liming Wu, eds.), Science Press, Beijing, 1997, pp. 85-138. · Zbl 0931.41017 |

[21] | Jiangyan Pu and Qi Zhang, Dynamic programming principle for stochastic recursive control problem with non-Lipschitz aggregator and associated Hamilton-Jacobi-Bellman equation, ESAIM Control Optim. Calc. Var. 24 (2018), no. 1, 355-376. · Zbl 1396.93135 |

[22] | Lishun Xiao and Shengjun Fan, A representation theorem for generators of BSDEs with general growth generators in \(y\) and its applications, Statist. Probab. Lett. 129 (2017), 297-305. · Zbl 1386.60207 |

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