# American Institute of Mathematical Sciences

November  2015, 35(11): 5521-5553. doi: 10.3934/dcds.2015.35.5521

## Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations

 1 Institute of Mathematical Finance and Department of Finance and Control Sciences, School of Mathematical Sciences, Fudan University, Shanghai 200433, China, China

Received  December 2013 Revised  October 2014 Published  May 2015

In this paper we study the optimal stochastic control problem for a path-dependent stochastic system under a recursive path-dependent cost functional, whose associated Bellman equation from dynamic programming principle is a path-dependent fully nonlinear partial differential equation of second order. A novel notion of viscosity solutions is introduced by restricting the semi-jets on an $\alpha$-Hölder space $\mathbf{C}^{\alpha}$ for $\alpha\in(0,\frac{1}{2})$. Using Dupire's functional Itô calculus, we prove that the value functional of the optimal stochastic control problem is a viscosity solution to the associated path-dependent Bellman equation. A state-dependent approximation of the path-dependent value functional is given.
Citation: Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521
##### References:
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Krylov, Control of the solution of a stochastic integral equation,, Teor. Verojatnost. i Primenen., 17 (1972), 111. Google Scholar [27] P.-L. Lions, 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 (1988), 243. doi: 10.1007/BF02392299. Google Scholar [28] P.-L. Lions, 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, (1988), 147. doi: 10.1007/BFb0083943. Google Scholar [29] P.-L. Lions and P. E. Souganidis, Fully nonlinear stochastic partial differential equations,, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 1085. doi: 10.1016/S0764-4442(98)80067-0. Google Scholar [30] P.-L. Lions and P. E. Souganidis, Fully nonlinear stochastic partial differential equations: Non-smooth equations and applications,, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 735. doi: 10.1016/S0764-4442(98)80161-4. Google Scholar [31] N. Y. Lukoyanov, On viscosity solution of functional Hamilton-Jacobi type equations for hereditary systems,, Proceedings of the Steklov Institute of Mathematics, 259 (2007). doi: 10.1134/S0081543807060132. Google Scholar [32] J. Ma, P. Protter and J. M. Yong, Solving forward-backward stochastic differential equations explicitly-A four step scheme,, Probab. Theory Related Fields, 98 (1994), 339. doi: 10.1007/BF01192258. Google Scholar [33] J. Ma and J. Yong, Adapted solution of a degenerate backward SPDE, with applications,, Stochastic Process. Appl., 70 (1997), 59. doi: 10.1016/S0304-4149(97)00057-4. Google Scholar [34] E. Pardoux and S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations,, in Stochastic Partial Differential Equations and their Applications (Charlotte, (1991), 200. doi: 10.1007/BFb0007334. Google Scholar [35] E. Pardoux and S. Tang, Forward-backward stochastic differential equations and quasilinear parabolic PDEs,, Probab. Theory Related Fields, 114 (1999), 123. doi: 10.1007/s004409970001. Google Scholar [36] S. Peng and F. Wang, BSDE, Path-dependent PDE and Nonlinear Feynman-Kac Formula,, preprint, (). Google Scholar [37] S. Peng, Stochastic Hamilton-Jacobi-Bellman equations,, SIAM J. Control Optim., 30 (1992), 284. doi: 10.1137/0330018. Google Scholar [38] S. Peng, BSDE and stochastic optimizations,, in Topics in Stochastic Analysis, (1997). Google Scholar [39] S. Peng, Backward stochastic differential equation, nonlinear expectation and their applications,, in Proceedings of the International Congress of Mathematicians. Vol. I, (2010), 393. Google Scholar [40] T. Pham and J. Zhang, Two person zero-sum game in weak formulation and path dependent Bellman-Isaacs equation,, SIAM J. Control Optim., 52 (2014), 2090. doi: 10.1137/120894907. Google Scholar [41] J. Qiu and S. Tang, Maximum principle for quasi-linear backward stochastic partial differential equations,, J. Funct. Anal., 262 (2012), 2436. doi: 10.1016/j.jfa.2011.12.002. Google Scholar [42] H. M. Soner, N. Touzi and J. Zhang, Wellposedness of second order backward SDEs,, Probab. Theory Related Fields, 153 (2012), 149. doi: 10.1007/s00440-011-0342-y. Google Scholar [43] D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes,, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1979). Google Scholar [44] S. Tang, Semi-linear systems of backward stochastic partial differential equations in $\mathbbR^n$,, Chinese Ann. Math. Ser. B, 26 (2005), 437. doi: 10.1142/S025295990500035X. Google Scholar [45] S. Tang, Dual representation as stochastic differential games of backward stochastic differential equations and dynamic evaluations,, C. R. Math. Acad. Sci. Paris, 342 (2006), 773. doi: 10.1016/j.crma.2006.03.025. Google Scholar [46] J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Vol. 43,, Springer Verlag, (1999). doi: 10.1007/978-1-4612-1466-3. Google Scholar [47] X. Y. Zhou, A duality analysis on stochastic partial differential equations,, J. Funct. Anal., 103 (1992), 275. doi: 10.1016/0022-1236(92)90122-Y. Google Scholar

show all references

##### References:
 [1] B. Boufoussi, J. Van Casteren and N. Mrhardy, Generalized backward doubly stochastic differential equations and SPDEs with nonlinear Neumann boundary conditions,, Bernoulli, 13 (2007), 423. doi: 10.3150/07-BEJ5092. Google Scholar [2] R. Buckdahn and J. Ma, Stochastic viscosity solutions for nonlinear stochastic partial differential equations. I,, Stochastic Process. Appl., 93 (2001), 181. doi: 10.1016/S0304-4149(00)00093-4. Google Scholar [3] R. Buckdahn and J. Ma, Stochastic viscosity solutions for nonlinear stochastic partial differential equations. II,, Stochastic Process. Appl., 93 (2001), 205. doi: 10.1016/S0304-4149(00)00092-2. Google Scholar [4] R. Buckdahn and J. Ma, Pathwise stochastic Taylor expansions and stochastic viscosity solutions for fully nonlinear stochastic PDEs,, Ann. Probab., 30 (2002), 1131. doi: 10.1214/aop/1029867123. Google Scholar [5] R. Buckdahn and J. Ma, Pathwise stochastic control problems and stochastic HJB equations,, SIAM J. Control Optim., 45 (2007), 2224. doi: 10.1137/S036301290444335X. Google Scholar [6] P. Cheridito, H. M. Soner, N. Touzi and N. Victoir, Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs,, Comm. Pure Appl. Math., 60 (2007), 1081. doi: 10.1002/cpa.20168. Google Scholar [7] R. Cont and D. A. Fournie, Functional Itô calculus and stochastic integral representation of martingales,, Ann. Probab., 41 (2013), 109. doi: 10.1214/11-AOP721. Google Scholar [8] R. Cont and D.-A. Fournié, Change of variable formulas for non-anticipative functionals on path space,, J. Funct. Anal., 259 (2010), 1043. doi: 10.1016/j.jfa.2010.04.017. Google Scholar [9] M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations,, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1. doi: 10.1090/S0273-0979-1992-00266-5. Google Scholar [10] M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations,, Trans. Amer. Math. Soc., 277 (1983), 1. doi: 10.1090/S0002-9947-1983-0690039-8. Google Scholar [11] K. Du and Q. Meng, A revisit to $W^n_2$-theory of super-parabolic backward stochastic partial differential equations in $\mathbb R^d$,, Stochastic Process. Appl., 120 (2010), 1996. doi: 10.1016/j.spa.2010.06.001. Google Scholar [12] K. Du and S. Tang, Strong solution of backward stochastic partial differential equations in $C^2$ domains,, Probab. Theory Related Fields, 154 (2012), 255. doi: 10.1007/s00440-011-0369-0. Google Scholar [13] K. Du, S. Tang and Q. Zhang, $W^{m,p}$-solution $(p\geq 2)$ of linear degenerate backward stochastic partial differential equations in the whole space,, J. Differential Equations, 254 (2013), 2877. doi: 10.1016/j.jde.2013.01.013. Google Scholar [14] B. Dupire, Functional Itô Calculus,, Bloomberg Portfolio Research paper No. 2009-04-FRONTIERS, (2009), 2009. Google Scholar [15] I. Ekren, N. Touzi and J. Zhang, Optimal stopping under nonlinear expectation,, Stochastic Process. Appl., 124 (2014), 3277. doi: 10.1016/j.spa.2014.04.006. Google Scholar [16] I. Ekren, N. Touzi and J. Zhang, Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part I,, preprint, (). Google Scholar [17] I. Ekren, N. Touzi and J. Zhang, Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II,, preprint, (). Google Scholar [18] I. Ekren, C. Keller, N. Touzi and J. Zhang, On viscosity solutions of path dependent PDEs,, Ann. Probab., 42 (2014), 204. doi: 10.1214/12-AOP788. Google Scholar [19] N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance,, Math. Finance, 7 (1997), 1. doi: 10.1111/1467-9965.00022. Google Scholar [20] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions,, 2nd edition, (2006). Google Scholar [21] M. Fuhrman, F. Masiero and G. Tessitore, Stochastic equations with delay: Optimal control via BSDEs and regular solutions of Hamilton-Jacobi-Bellman equations,, SIAM J. Control Optim., 48 (2010), 4624. doi: 10.1137/080730354. Google Scholar [22] B. Goldys and F. Gozzi, Second order parabolic Hamilton-Jacobi-Bellman equations in Hilbert spaces and stochastic control: $L_\mu^2$ approach,, Stochastic Process. Appl., 116 (2006), 1932. doi: 10.1016/j.spa.2006.05.006. Google Scholar [23] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus,, 2nd edition, (1991). doi: 10.1007/978-1-4612-0949-2. Google Scholar [24] I. Karatzas and S. E. Shreve, Methods of Mathematical Finance,, Applications of Mathematics (New York), (1998). doi: 10.1007/b98840. Google Scholar [25] A. V. Kim, Functional Differential Equations. Application of $i$-Smooth Calculus,, Mathematics and its Applications, (1999). doi: 10.1007/978-94-017-1630-7. Google Scholar [26] N. V. Krylov, Control of the solution of a stochastic integral equation,, Teor. Verojatnost. i Primenen., 17 (1972), 111. Google Scholar [27] P.-L. Lions, 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 (1988), 243. doi: 10.1007/BF02392299. Google Scholar [28] P.-L. Lions, 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, (1988), 147. doi: 10.1007/BFb0083943. Google Scholar [29] P.-L. Lions and P. E. Souganidis, Fully nonlinear stochastic partial differential equations,, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 1085. doi: 10.1016/S0764-4442(98)80067-0. Google Scholar [30] P.-L. Lions and P. E. Souganidis, Fully nonlinear stochastic partial differential equations: Non-smooth equations and applications,, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 735. doi: 10.1016/S0764-4442(98)80161-4. Google Scholar [31] N. Y. Lukoyanov, On viscosity solution of functional Hamilton-Jacobi type equations for hereditary systems,, Proceedings of the Steklov Institute of Mathematics, 259 (2007). doi: 10.1134/S0081543807060132. Google Scholar [32] J. Ma, P. Protter and J. M. Yong, Solving forward-backward stochastic differential equations explicitly-A four step scheme,, Probab. Theory Related Fields, 98 (1994), 339. doi: 10.1007/BF01192258. Google Scholar [33] J. Ma and J. Yong, Adapted solution of a degenerate backward SPDE, with applications,, Stochastic Process. Appl., 70 (1997), 59. doi: 10.1016/S0304-4149(97)00057-4. Google Scholar [34] E. Pardoux and S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations,, in Stochastic Partial Differential Equations and their Applications (Charlotte, (1991), 200. doi: 10.1007/BFb0007334. Google Scholar [35] E. Pardoux and S. Tang, Forward-backward stochastic differential equations and quasilinear parabolic PDEs,, Probab. Theory Related Fields, 114 (1999), 123. doi: 10.1007/s004409970001. Google Scholar [36] S. Peng and F. Wang, BSDE, Path-dependent PDE and Nonlinear Feynman-Kac Formula,, preprint, (). Google Scholar [37] S. Peng, Stochastic Hamilton-Jacobi-Bellman equations,, SIAM J. Control Optim., 30 (1992), 284. doi: 10.1137/0330018. Google Scholar [38] S. Peng, BSDE and stochastic optimizations,, in Topics in Stochastic Analysis, (1997). Google Scholar [39] S. Peng, Backward stochastic differential equation, nonlinear expectation and their applications,, in Proceedings of the International Congress of Mathematicians. Vol. I, (2010), 393. Google Scholar [40] T. Pham and J. Zhang, Two person zero-sum game in weak formulation and path dependent Bellman-Isaacs equation,, SIAM J. Control Optim., 52 (2014), 2090. doi: 10.1137/120894907. Google Scholar [41] J. Qiu and S. Tang, Maximum principle for quasi-linear backward stochastic partial differential equations,, J. Funct. Anal., 262 (2012), 2436. doi: 10.1016/j.jfa.2011.12.002. Google Scholar [42] H. M. Soner, N. Touzi and J. Zhang, Wellposedness of second order backward SDEs,, Probab. Theory Related Fields, 153 (2012), 149. doi: 10.1007/s00440-011-0342-y. Google Scholar [43] D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes,, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1979). Google Scholar [44] S. Tang, Semi-linear systems of backward stochastic partial differential equations in $\mathbbR^n$,, Chinese Ann. Math. Ser. B, 26 (2005), 437. doi: 10.1142/S025295990500035X. Google Scholar [45] S. Tang, Dual representation as stochastic differential games of backward stochastic differential equations and dynamic evaluations,, C. R. Math. Acad. Sci. Paris, 342 (2006), 773. doi: 10.1016/j.crma.2006.03.025. Google Scholar [46] J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Vol. 43,, Springer Verlag, (1999). doi: 10.1007/978-1-4612-1466-3. Google Scholar [47] X. Y. Zhou, A duality analysis on stochastic partial differential equations,, J. Funct. Anal., 103 (1992), 275. doi: 10.1016/0022-1236(92)90122-Y. Google Scholar
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