# American Institute of Mathematical Sciences

January  2016, 1: 6 doi: 10.1186/s41546-016-0010-3

## Pseudo-Markovian viscosity solutions of fully nonlinear degenerate PPDEs

 1 ETH Department of Mathematics, Zurich, Switzerland; 2 University of Southern California, Department of Mathematics, Los Angeles, California, USA

Received  April 07, 2016 Revised  August 07, 2016 Published  December 2016

Fund Project: Research supported in part by NSF grant DMS 1413717.

In this paper, we propose a new type of viscosity solutions for fully nonlinear path-dependent PDEs. By restricting the solution to a pseudo-Markovian structure defined below, we remove the uniform non-degeneracy condition needed in our earlier works (Ekren, I, Touzi, N, Zhang, J, Ann Probab, 44:1212-1253, 2016a; Ekren, I, Touzi, N, Zhang, J, Ann Probab, 44:2507-2553, 2016b) to establish the uniqueness result. We establish the comparison principle under natural and mild conditions. Moreover, we apply our results to two important classes of PPDEs:the stochastic HJB equations and the path-dependent Isaacs equations, induced from the stochastic optimization with random coefficients and the path-dependent zero-sum game problem, respectively.
Citation: Ibrahim Ekren, Jianfeng Zhang. Pseudo-Markovian viscosity solutions of fully nonlinear degenerate PPDEs. Probability, Uncertainty and Quantitative Risk, 2016, 1 (0) : 6-. doi: 10.1186/s41546-016-0010-3
##### References:
 [1] Barles, G, Buckdahn, R, Pardoux, E:Backward stochastic differential equations and integral-partial differential equations. Stoch. Stoch. Rep 60, 57-83 (1997) [2] Bayraktar, E, Yao, S:Optimal Stopping with Random Maturity under Nonlinear Expectations. preprint(2016). arXiv:1505.07533 [3] Cont, R, Fournie, D:Functional Itô calculus and stochastic integral representation of martingales. Ann. Probab 41, 109-133 (2013) [4] Cosso, A, Russo, F:Strong-viscosity solutions:Semilinear parabolic PDEs and path-dependent PDEs.preprint (2016). arXiv:1505.02927 [5] Crandall, MG, Ishii, H, Lions, P-L:User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (NS) 27, 1-67 (1992) [6] Dupire, B:Functional Itô calculus (2009). papers.ssrn.com [7] Ekren, I, Keller, C, Touzi, N, Zhang, J:On Viscosity Solutions of Path Dependent PDEs. Ann. Probab 42, 204-236 (2014a) [8] Ekren, I, Touzi, N, Zhang, J:Optimal Stopping under Nonlinear Expectation. Stochastic Process. Appl 124, 3277-3311 (2014b) [9] Ekren, I, Touzi, N, Zhang, J:Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs:Part I. Ann. Probab 44, 1212-1253 (2016a) [10] Ekren, I, Touzi, N, Zhang, J:Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs:Part II. Ann. Probab 44, 2507-2553 (2016b) [11] Fleming, W, Soner, HM Controlled Markov Processes and Viscosity Solutions, 2nd ed. Springer, New York (2006) [12] Fleming, W, Souganidis, PE:On The Existence of Value Functions of Two-Player, Zero-Sum Stochastic Differential Games. Indiana Univ. Math. J 38, 293-314 (1989) [13] Mikulevicious, R:On the convergence of diffusions, Stochastic Differential Systems, pp. 176-186.Springer-Verlag, Berlin (1987) [14] Mikulevicius, R, Rozovskii, B:Martingale problems for stochastic PDE's. Stochastic partial differential equations:six perspectives, pp. 243-325 (1999). Math. Surveys Monogr., 64, Amer. Math. Soc., Providence, RI Peng, S:Stochastic Hamilton-Jacobi-Bellman Equations. SIAM J. Control Optim 30, 284-304 (1992) [15] Peng, S:Open problems on backward stochastic differential equations. Control of distributed parameter and stochastic systems, pp. 265-273. Springer, US (1999) [16] Peng, S:Backward stochastic differential equation, nonlinear expectation and their applications. Proceedings of the International Congress of Mathematicians, Hyderabad, India (2010) [17] Peng, S:Note on Viscosity Solution of Path-Dependent PDE and G-Martingales. preprint, arXiv:1106.1144 (2011) [18] Peng, S, Song, Y:G-Expectation Weighted Sobolev Spaces, Backward SDE and Path Dependent PDE. J.Math. Soc. Japan 67, 1725-1757 (2015) [19] Peng, S, Wang, F:BSDE, Path-dependent PDE and Nonlinear Feynman-Kac Formula. Sci. China Math 59, 19-36 (2016) [20] Pham, T, Zhang, J:Some Norm Estimates for Semimartingales. Electron. J. Probab 18, 1-25 (2013) [21] Pham, T, Zhang, J:Two Person Zero-sum Game in Weak Formulation and Path Dependent Bellman-Isaacs Equation. SIAM J. Control Optim 52, 2090-2121 (2014) [22] Qiu, J:Weak Solution for a Class of Fully Nonlinear Stochastic Hamilton-Jacobi-Bellman Equations.preprint (2016). arXiv:1410.6967 [23] Ren, Z:Perron's method for viscosity solutions of semilinear path dependent PDEs, Stochastics:An International Journal of Probability and Stochastic Processes (2016) [24] Ren, Z, Touzi, N, Zhang, J:An Overview of Viscosity Solutions of Path-Dependent PDEs. Stochastic Anal. Appl 100, 397-453 (2014) [25] Ren, Z, Touzi, N, Zhang, J:Comparison of Viscosity Solutions of Semilinear Path-Dependent PDEs, preprint (2016a). arXiv:1410.7281 [26] Ren, Z, Touzi, N, Zhang, J:Comparison of Viscosity Solutions of Fully Nonlinear Degenerate Parabolic Path-dependent PDEs, preprint (2016b). arXiv:1511.05910

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##### References:
 [1] Barles, G, Buckdahn, R, Pardoux, E:Backward stochastic differential equations and integral-partial differential equations. Stoch. Stoch. Rep 60, 57-83 (1997) [2] Bayraktar, E, Yao, S:Optimal Stopping with Random Maturity under Nonlinear Expectations. preprint(2016). arXiv:1505.07533 [3] Cont, R, Fournie, D:Functional Itô calculus and stochastic integral representation of martingales. Ann. Probab 41, 109-133 (2013) [4] Cosso, A, Russo, F:Strong-viscosity solutions:Semilinear parabolic PDEs and path-dependent PDEs.preprint (2016). arXiv:1505.02927 [5] Crandall, MG, Ishii, H, Lions, P-L:User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (NS) 27, 1-67 (1992) [6] Dupire, B:Functional Itô calculus (2009). papers.ssrn.com [7] Ekren, I, Keller, C, Touzi, N, Zhang, J:On Viscosity Solutions of Path Dependent PDEs. Ann. Probab 42, 204-236 (2014a) [8] Ekren, I, Touzi, N, Zhang, J:Optimal Stopping under Nonlinear Expectation. Stochastic Process. Appl 124, 3277-3311 (2014b) [9] Ekren, I, Touzi, N, Zhang, J:Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs:Part I. Ann. Probab 44, 1212-1253 (2016a) [10] Ekren, I, Touzi, N, Zhang, J:Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs:Part II. Ann. Probab 44, 2507-2553 (2016b) [11] Fleming, W, Soner, HM Controlled Markov Processes and Viscosity Solutions, 2nd ed. Springer, New York (2006) [12] Fleming, W, Souganidis, PE:On The Existence of Value Functions of Two-Player, Zero-Sum Stochastic Differential Games. Indiana Univ. Math. J 38, 293-314 (1989) [13] Mikulevicious, R:On the convergence of diffusions, Stochastic Differential Systems, pp. 176-186.Springer-Verlag, Berlin (1987) [14] Mikulevicius, R, Rozovskii, B:Martingale problems for stochastic PDE's. Stochastic partial differential equations:six perspectives, pp. 243-325 (1999). Math. Surveys Monogr., 64, Amer. Math. Soc., Providence, RI Peng, S:Stochastic Hamilton-Jacobi-Bellman Equations. SIAM J. Control Optim 30, 284-304 (1992) [15] Peng, S:Open problems on backward stochastic differential equations. Control of distributed parameter and stochastic systems, pp. 265-273. Springer, US (1999) [16] Peng, S:Backward stochastic differential equation, nonlinear expectation and their applications. Proceedings of the International Congress of Mathematicians, Hyderabad, India (2010) [17] Peng, S:Note on Viscosity Solution of Path-Dependent PDE and G-Martingales. preprint, arXiv:1106.1144 (2011) [18] Peng, S, Song, Y:G-Expectation Weighted Sobolev Spaces, Backward SDE and Path Dependent PDE. J.Math. Soc. Japan 67, 1725-1757 (2015) [19] Peng, S, Wang, F:BSDE, Path-dependent PDE and Nonlinear Feynman-Kac Formula. Sci. China Math 59, 19-36 (2016) [20] Pham, T, Zhang, J:Some Norm Estimates for Semimartingales. Electron. J. Probab 18, 1-25 (2013) [21] Pham, T, Zhang, J:Two Person Zero-sum Game in Weak Formulation and Path Dependent Bellman-Isaacs Equation. SIAM J. Control Optim 52, 2090-2121 (2014) [22] Qiu, J:Weak Solution for a Class of Fully Nonlinear Stochastic Hamilton-Jacobi-Bellman Equations.preprint (2016). arXiv:1410.6967 [23] Ren, Z:Perron's method for viscosity solutions of semilinear path dependent PDEs, Stochastics:An International Journal of Probability and Stochastic Processes (2016) [24] Ren, Z, Touzi, N, Zhang, J:An Overview of Viscosity Solutions of Path-Dependent PDEs. Stochastic Anal. Appl 100, 397-453 (2014) [25] Ren, Z, Touzi, N, Zhang, J:Comparison of Viscosity Solutions of Semilinear Path-Dependent PDEs, preprint (2016a). arXiv:1410.7281 [26] Ren, Z, Touzi, N, Zhang, J:Comparison of Viscosity Solutions of Fully Nonlinear Degenerate Parabolic Path-dependent PDEs, preprint (2016b). arXiv:1511.05910
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