# zbMATH — the first resource for mathematics

Viscosity solutions to parabolic master equations and McKean-Vlasov SDEs with closed-loop controls. (English) Zbl 1445.35118
Summary: The master equation is a type of PDE whose state variable involves the distribution of certain underlying state process. It is a powerful tool for studying the limit behavior of large interacting systems, including mean field games and systemic risk. It also appears naturally in stochastic control problems with partial information and in time inconsistent problems. In this paper we propose a novel notion of viscosity solution for parabolic master equations, arising mainly from control problems, and establish its wellposedness. Our main innovation is to restrict the involved measures to a certain set of semimartingale measures which satisfy the desired compactness. As an important example, we study the HJB master equation associated with the control problems for McKean-Vlasov SDEs. Due to practical considerations, we consider closed-loop controls. It turns out that the regularity of the value function becomes much more involved in this framework than the counterpart in the standard control problems. Finally, we build the whole theory in the path dependent setting, which is often seen in applications. The main result in this part is an extension of B. Dupire’s functional Itô formula [Quant. Finance 19, No. 5, 721–729 (2019; Zbl 1420.91458)]. This Itô formula requires a special structure of the derivatives with respect to the measures, which was originally due to Lions in the state dependent case. We provided an elementary proof for this well known result in the short note [“An elementary proof for the structure of Wasserstein derivatives”, Preprint, arXiv:1705.08046], and the same arguments work in the path dependent setting here.

##### MSC:
 35D40 Viscosity solutions to PDEs 35K55 Nonlinear parabolic equations 49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games 60H30 Applications of stochastic analysis (to PDEs, etc.) 35R15 PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables) 49L20 Dynamic programming in optimal control and differential games 93E20 Optimal stochastic control
Full Text:
##### References:
 [1] Bandini, E., Cosso, A., Fuhrman, M. and Pham, H. (2018). Backward SDEs for optimal control of partially observed path-dependent stochastic systems: A control randomization approach. Ann. Appl. Probab. 28 1634-1678. · Zbl 1431.60045 [2] Bandini, E., Cosso, A., Fuhrman, M. and Pham, H. (2019). Randomized filtering and Bellman equation in Wasserstein space for partial observation control problem. Stochastic Process. Appl. 129 674-711. · Zbl 1405.93229 [3] Bayraktar, E., Cosso, A. and Pham, H. (2018). Randomized dynamic programming principle and Feynman-Kac representation for optimal control of McKean-Vlasov dynamics. Trans. Amer. Math. Soc. 370 2115-2160. · Zbl 1381.93102 [4] Bensoussan, A., Frehse, J. and Yam, P. (2013). Mean Field Games and Mean Field Type Control Theory. Springer Briefs in Mathematics. Springer, New York. · Zbl 1287.93002 [5] Bensoussan, A., Graber, P. and Yam, S. C. P. Stochastic control on space of random variables. Preprint. Available at arXiv:1903.12602. [6] Bensoussan, A. and Yam, S. C. P. (2019). Control problem on space of random variables and master equation. ESAIM Control Optim. Calc. Var. 25 Art. 10. · Zbl 1450.35305 [7] Buckdahn, R., Li, J., Peng, S. and Rainer, C. (2017). Mean-field stochastic differential equations and associated PDEs. Ann. Probab. 45 824-878. · Zbl 1402.60070 [8] Cardaliaguet, P. (2013). Notes on mean field games. From P.-L. Lions Lectures at Coll‘ege de France. Available at http://www.college-de-france.fr. [9] Cardaliaguet, P., Delarue, F., Lasry, J.-M. and Lions, P.-L. (2019). The Master Equation and the Convergence Problem in Mean Field Games. Annals of Mathematics Studies 201. Princeton Univ. Press, Princeton, NJ. · Zbl 1430.91002 [10] Carmona, R. and Delarue, F. (2014). The master equation for large population equilibriums. In Stochastic Analysis and Applications 2014 (D. Crisan, B. Hambly and T. Zariphopoulou, eds.). Springer Proc. Math. Stat. 100 77-128. Springer, Cham. · Zbl 1391.92036 [11] Carmona, R. and Delarue, F. (2018). Probabilistic Theory of Mean Field Games with Applications. I: Mean Field FBSDEs, Control, and Games. Probability Theory and Stochastic Modelling 83. Springer, Cham. · Zbl 1422.91014 [12] Carmona, R. and Delarue, F. (2018). Probabilistic Theory of Mean Field Games with Applications. II: Mean Field Games with Common Noise and Master Equations. Probability Theory and Stochastic Modelling 84. Springer, Cham. · Zbl 1422.91015 [13] Chassagneux, J.-F., Crisan, D. and Delarue, F. A probabilistic approach to classical solutions of the master equation for large population equilibria. Preprint. Available at arXiv:1411.3009. [14] Cont, R. and Fournié, D.-A. (2013). Functional Itô calculus and stochastic integral representation of martingales. Ann. Probab. 41 109-133. · Zbl 1272.60031 [15] Cosso, A. and Pham, H. (2019). Zero-sum stochastic differential games of generalized McKean-Vlasov type. J. Math. Pures Appl. (9) 129 180-212. Available at arXiv:1803.07329. · Zbl 1423.49039 [16] Dupire, B. (2009). Functional Itô calculus. Available at http://ssrn.com/abstract=1435551. · Zbl 1420.91458 [17] Ekren, I., Keller, C., Touzi, N. and Zhang, J. (2014). On viscosity solutions of path dependent PDEs. Ann. Probab. 42 204-236. · Zbl 1320.35154 [18] Ekren, I., Touzi, N. and Zhang, J. (2016). Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part I. Ann. Probab. 44 1212-1253. · Zbl 1375.35250 [19] Ekren, I., Touzi, N. and Zhang, J. (2016). Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II. Ann. Probab. 44 2507-2553. · Zbl 1394.35228 [20] Fabbri, G., Gozzi, F. and Świçch, A. (2017). Stochastic Optimal Control in Infinite Dimension: Dynamic Programming and HJB Equations. Probability Theory and Stochastic Modelling 82. Springer, Cham. [21] Gangbo, W. and Świçch, A. (2015). Metric viscosity solutions of Hamilton-Jacobi equations depending on local slopes. Calc. Var. Partial Differential Equations 54 1183-1218. · Zbl 1355.49024 [22] Gangbo, W. and Świçch, A. (2015). Existence of a solution to an equation arising from the theory of mean field games. J. Differential Equations 259 6573-6643. · Zbl 1359.35221 [23] Gangbo, W. and Tudorascu, A. (2019). On differentiability in the Wasserstein space and well-posedness for Hamilton-Jacobi equations. J. Math. Pures Appl. (9) 125 119-174. · Zbl 1419.35234 [24] Huang, M., Malhamé, R. P. and Caines, P. E. (2006). Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6 221-251. · Zbl 1136.91349 [25] Lacker, D. (2017). Limit theory for controlled McKean-Vlasov dynamics. SIAM J. Control Optim. 55 1641-1672. · Zbl 1362.93167 [26] Lasry, J.-M. and Lions, P.-L. (2007). Mean field games. Jpn. J. Math. 2 229-260. · Zbl 1156.91321 [27] Lions, P.-L. (1988). Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. I. The case of bounded stochastic evolutions. Acta Math. 161 243-278. · Zbl 0757.93082 [28] Lions, P.-L. (1989). Viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions. II. Optimal control of Zakai’s equation. In Stochastic Partial Differential Equations and Applications, II (Trento, 1988). Lecture Notes in Math. 1390 147-170. Springer, Berlin. · Zbl 0757.93083 [29] Lions, P.-L. (1989). Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. III. Uniqueness of viscosity solutions for general second-order equations. J. Funct. Anal. 86 1-18. · Zbl 0757.93084 [30] Lions, P. L. Cours au College de France. Available at www.college-de-france.fr. [31] Ma, J., Wong, L. and Zhang, J. (2018). Time consistent conditional expectation under probability distortion. Preprint. Available at arXiv:1809.08262. [32] Mou, C. and Zhang, J. (2019). Weak solutions of mean field game master equations. Preprint. Available at arXiv:1903.09907. [33] Peng, S. and Wang, F. (2016). BSDE, path-dependent PDE and nonlinear Feynman-Kac formula. Sci. China Math. 59 19-36. · Zbl 1342.60108 [34] Pham, H. and Wei, X. (2018). Bellman equation and viscosity solutions for mean-field stochastic control problem. ESAIM Control Optim. Calc. Var. 24 437-461. · Zbl 1396.93134 [35] Pham, T. and Zhang, J. (2014). Two person zero-sum game in weak formulation and path dependent Bellman-Isaacs equation. SIAM J. Control Optim. 52 2090-2121. · Zbl 1308.91029 [36] Possamai, D., Touzi, N. and Zhang, J. (2018). Zero-sum path-dependent stochastic differential games in weak formulation. Ann. Appl. Probab. To appear. Available at arXiv:1808.03756. [37] Ren, Z. and Rosestolato, M. Viscosity solutions of path-dependent PDEs with randomized time. (2018). Preprint. Available at arXiv:1806.07654. · Zbl 1445.35207 [38] Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin. · Zbl 0731.60002 [39] Saporito, Y. F. and Zhang, J. (2019). Stochastic control with delayed information and related nonlinear master equation. SIAM J. Control Optim. 57 693-717. · Zbl 1418.60096 [40] Sîrbu, M. (2014). Stochastic Perron’s method and elementary strategies for zero-sum differential games. SIAM J. Control Optim. 52 1693-1711. · Zbl 1409.91029 [41] Viens, F. and Zhang, J. (2019). A martingale approach for fractional Brownian motions and related path dependent PDEs. Ann. Appl. Probab. 29 3489-3540. · Zbl 1441.60031 [42] Wu, C. and Zhang, J. (2017). An elementary proof for the structure of Wasserstein derivatives. Preprint. Available at arXiv:1705.08046. [43] Zhang, J. (2017). Backward Stochastic Differential Equations: From Linear to Fully Nonlinear Theory. Probability Theory and Stochastic Modelling 86. Springer, New York. [44] Zheng, W. A. (1985). Tightness results for laws of diffusion processes application to stochastic mechanics. Ann. Inst. Henri Poincaré Probab. Stat. 21 103-124. · Zbl 0579.60050 [45] Zhou, X. · Zbl 1225.91062
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.