American Institute of Mathematical Sciences

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

Pseudo-Markovian viscosity solutions of fully nonlinear degenerate PPDEs

Ibrahim Ekren 1,, and  Jianfeng Zhang 2,,

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.
Keywords: Path dependent PDEs, Viscosity solutions, Comparison principle, Stochastic HJB equations, Isaacs equations.
Mathematics Subject Classification: 35D40;35K10;60H10;60H30.
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) Google Scholar

[2]

Bayraktar, E, Yao, S:Optimal Stopping with Random Maturity under Nonlinear Expectations. preprint(2016). arXiv:1505.07533 Google Scholar

[3]

Cont, R, Fournie, D:Functional Itô calculus and stochastic integral representation of martingales. Ann. Probab 41, 109-133 (2013) Google Scholar

[4]

Cosso, A, Russo, F:Strong-viscosity solutions:Semilinear parabolic PDEs and path-dependent PDEs.preprint (2016). arXiv:1505.02927 Google Scholar

[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) Google Scholar

[6]

Dupire, B:Functional Itô calculus (2009). papers.ssrn.com Google Scholar

[7]

Ekren, I, Keller, C, Touzi, N, Zhang, J:On Viscosity Solutions of Path Dependent PDEs. Ann. Probab 42, 204-236 (2014a) Google Scholar

[8]

Ekren, I, Touzi, N, Zhang, J:Optimal Stopping under Nonlinear Expectation. Stochastic Process. Appl 124, 3277-3311 (2014b) Google Scholar

[9]

Ekren, I, Touzi, N, Zhang, J:Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs:Part I. Ann. Probab 44, 1212-1253 (2016a) Google Scholar

[10]

Ekren, I, Touzi, N, Zhang, J:Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs:Part II. Ann. Probab 44, 2507-2553 (2016b) Google Scholar

[11]

Fleming, W, Soner, HM Controlled Markov Processes and Viscosity Solutions, 2nd ed. Springer, New York (2006) Google Scholar

[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) Google Scholar

[13]

Mikulevicious, R:On the convergence of diffusions, Stochastic Differential Systems, pp. 176-186.Springer-Verlag, Berlin (1987) Google Scholar

[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) Google Scholar

[15]

Peng, S:Open problems on backward stochastic differential equations. Control of distributed parameter and stochastic systems, pp. 265-273. Springer, US (1999) Google Scholar

[16]

Peng, S:Backward stochastic differential equation, nonlinear expectation and their applications. Proceedings of the International Congress of Mathematicians, Hyderabad, India (2010) Google Scholar

[17]

Peng, S:Note on Viscosity Solution of Path-Dependent PDE and G-Martingales. preprint, arXiv:1106.1144 (2011) Google Scholar

[18]

Peng, S, Song, Y:G-Expectation Weighted Sobolev Spaces, Backward SDE and Path Dependent PDE. J.Math. Soc. Japan 67, 1725-1757 (2015) Google Scholar

[19]

Peng, S, Wang, F:BSDE, Path-dependent PDE and Nonlinear Feynman-Kac Formula. Sci. China Math 59, 19-36 (2016) Google Scholar

[20]

Pham, T, Zhang, J:Some Norm Estimates for Semimartingales. Electron. J. Probab 18, 1-25 (2013) Google Scholar

[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) Google Scholar

[22]

Qiu, J:Weak Solution for a Class of Fully Nonlinear Stochastic Hamilton-Jacobi-Bellman Equations.preprint (2016). arXiv:1410.6967 Google Scholar

[23]

Ren, Z:Perron's method for viscosity solutions of semilinear path dependent PDEs, Stochastics:An International Journal of Probability and Stochastic Processes (2016) Google Scholar

[24]

Ren, Z, Touzi, N, Zhang, J:An Overview of Viscosity Solutions of Path-Dependent PDEs. Stochastic Anal. Appl 100, 397-453 (2014) Google Scholar

[25]

Ren, Z, Touzi, N, Zhang, J:Comparison of Viscosity Solutions of Semilinear Path-Dependent PDEs, preprint (2016a). arXiv:1410.7281 Google Scholar

[26]

Ren, Z, Touzi, N, Zhang, J:Comparison of Viscosity Solutions of Fully Nonlinear Degenerate Parabolic Path-dependent PDEs, preprint (2016b). arXiv:1511.05910 Google Scholar

show all references

References:
[1]

Barles, G, Buckdahn, R, Pardoux, E:Backward stochastic differential equations and integral-partial differential equations. Stoch. Stoch. Rep 60, 57-83 (1997) Google Scholar

[2]

Bayraktar, E, Yao, S:Optimal Stopping with Random Maturity under Nonlinear Expectations. preprint(2016). arXiv:1505.07533 Google Scholar

[3]

Cont, R, Fournie, D:Functional Itô calculus and stochastic integral representation of martingales. Ann. Probab 41, 109-133 (2013) Google Scholar

[4]

Cosso, A, Russo, F:Strong-viscosity solutions:Semilinear parabolic PDEs and path-dependent PDEs.preprint (2016). arXiv:1505.02927 Google Scholar

[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) Google Scholar

[6]

Dupire, B:Functional Itô calculus (2009). papers.ssrn.com Google Scholar

[7]

Ekren, I, Keller, C, Touzi, N, Zhang, J:On Viscosity Solutions of Path Dependent PDEs. Ann. Probab 42, 204-236 (2014a) Google Scholar

[8]

Ekren, I, Touzi, N, Zhang, J:Optimal Stopping under Nonlinear Expectation. Stochastic Process. Appl 124, 3277-3311 (2014b) Google Scholar

[9]

Ekren, I, Touzi, N, Zhang, J:Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs:Part I. Ann. Probab 44, 1212-1253 (2016a) Google Scholar

[10]

Ekren, I, Touzi, N, Zhang, J:Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs:Part II. Ann. Probab 44, 2507-2553 (2016b) Google Scholar

[11]

Fleming, W, Soner, HM Controlled Markov Processes and Viscosity Solutions, 2nd ed. Springer, New York (2006) Google Scholar

[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) Google Scholar

[13]

Mikulevicious, R:On the convergence of diffusions, Stochastic Differential Systems, pp. 176-186.Springer-Verlag, Berlin (1987) Google Scholar

[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) Google Scholar

[15]

Peng, S:Open problems on backward stochastic differential equations. Control of distributed parameter and stochastic systems, pp. 265-273. Springer, US (1999) Google Scholar

[16]

Peng, S:Backward stochastic differential equation, nonlinear expectation and their applications. Proceedings of the International Congress of Mathematicians, Hyderabad, India (2010) Google Scholar

[17]

Peng, S:Note on Viscosity Solution of Path-Dependent PDE and G-Martingales. preprint, arXiv:1106.1144 (2011) Google Scholar

[18]

Peng, S, Song, Y:G-Expectation Weighted Sobolev Spaces, Backward SDE and Path Dependent PDE. J.Math. Soc. Japan 67, 1725-1757 (2015) Google Scholar

[19]

Peng, S, Wang, F:BSDE, Path-dependent PDE and Nonlinear Feynman-Kac Formula. Sci. China Math 59, 19-36 (2016) Google Scholar

[20]

Pham, T, Zhang, J:Some Norm Estimates for Semimartingales. Electron. J. Probab 18, 1-25 (2013) Google Scholar

[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) Google Scholar

[22]

Qiu, J:Weak Solution for a Class of Fully Nonlinear Stochastic Hamilton-Jacobi-Bellman Equations.preprint (2016). arXiv:1410.6967 Google Scholar

[23]

Ren, Z:Perron's method for viscosity solutions of semilinear path dependent PDEs, Stochastics:An International Journal of Probability and Stochastic Processes (2016) Google Scholar

[24]

Ren, Z, Touzi, N, Zhang, J:An Overview of Viscosity Solutions of Path-Dependent PDEs. Stochastic Anal. Appl 100, 397-453 (2014) Google Scholar

[25]

Ren, Z, Touzi, N, Zhang, J:Comparison of Viscosity Solutions of Semilinear Path-Dependent PDEs, preprint (2016a). arXiv:1410.7281 Google Scholar

[26]

Ren, Z, Touzi, N, Zhang, J:Comparison of Viscosity Solutions of Fully Nonlinear Degenerate Parabolic Path-dependent PDEs, preprint (2016b). arXiv:1511.05910 Google Scholar

[1]

Ludger Overbeck, Jasmin A. L. Röder. Path-dependent backward stochastic Volterra integral equations with jumps, differentiability and duality principle. Probability, Uncertainty and Quantitative Risk, 2018, 3 (0) : 4-. doi: 10.1186/s41546-018-0030-2
[2]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521
[3]

N. V. Krylov. Uniqueness for Lp-viscosity solutions for uniformly parabolic Isaacs equations with measurable lower order terms. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2495-2516. doi: 10.3934/cpaa.2018119
[4]

Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395
[5]

Rainer Buckdahn, Christian Keller, Jin Ma, Jianfeng Zhang. Fully nonlinear stochastic and rough PDEs: Classical and viscosity solutions. Probability, Uncertainty and Quantitative Risk, 2020, 5 (0) : 7-. doi: 10.1186/s41546-020-00049-8
[6]

Pablo Ochoa, Julio Alejo Ruiz. A study of comparison, existence and regularity of viscosity and weak solutions for quasilinear equations in the Heisenberg group. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1091-1115. doi: 10.3934/cpaa.2019053
[7]

Shigeaki Koike, Andrzej Świech. Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1897-1910. doi: 10.3934/cpaa.2012.11.1897
[8]

Thomas Leroy. Relativistic transfer equations: Comparison principle and convergence to the non-equilibrium regime. Kinetic & Related Models, 2015, 8 (4) : 725-763. doi: 10.3934/krm.2015.8.725
[9]

Maria Francesca Betta, Rosaria Di Nardo, Anna Mercaldo, Adamaria Perrotta. Gradient estimates and comparison principle for some nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2015, 14 (3) : 897-922. doi: 10.3934/cpaa.2015.14.897
[10]

Xing Huang, Michael Röckner, Feng-Yu Wang. Nonlinear Fokker–Planck equations for probability measures on path space and path-distribution dependent SDEs. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3017-3035. doi: 10.3934/dcds.2019125
[11]

Xulong Qin, Zheng-An Yao. Global solutions of the free boundary problem for the compressible Navier-Stokes equations with density-dependent viscosity. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1041-1052. doi: 10.3934/cpaa.2010.9.1041
[12]

Imran H. Biswas, Indranil Chowdhury. On the differentiability of the solutions of non-local Isaacs equations involving $\frac{1}{2}$-Laplacian. Communications on Pure & Applied Analysis, 2016, 15 (3) : 907-927. doi: 10.3934/cpaa.2016.15.907
[13]

Monica Motta, Caterina Sartori. Uniqueness of solutions for second order Bellman-Isaacs equations with mixed boundary conditions. Discrete & Continuous Dynamical Systems, 2008, 20 (4) : 739-765. doi: 10.3934/dcds.2008.20.739
[14]

Sergey E. Mikhailov, Carlos F. Portillo. Boundary-Domain Integral Equations equivalent to an exterior mixed BVP for the variable-viscosity compressible Stokes PDEs. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1103-1133. doi: 10.3934/cpaa.2021009
[15]

David Cheban, Zhenxin Liu. Averaging principle on infinite intervals for stochastic ordinary differential equations. Electronic Research Archive, 2021, 29 (4) : 2791-2817. doi: 10.3934/era.2021014
[16]

Martino Bardi, Gabriele Terrone. On the homogenization of some non-coercive Hamilton--Jacobi--Isaacs equations. Communications on Pure & Applied Analysis, 2013, 12 (1) : 207-236. doi: 10.3934/cpaa.2013.12.207
[17]

Huijie Qiao, Jiang-Lun Wu. On the path-independence of the Girsanov transformation for stochastic evolution equations with jumps in Hilbert spaces. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1449-1467. doi: 10.3934/dcdsb.2018215
[18]

Jeffrey R. L. Webb. Positive solutions of nonlinear equations via comparison with linear operators. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5507-5519. doi: 10.3934/dcds.2013.33.5507
[19]

Juan Li, Wenqiang Li. Controlled reflected mean-field backward stochastic differential equations coupled with value function and related PDEs. Mathematical Control & Related Fields, 2015, 5 (3) : 501-516. doi: 10.3934/mcrf.2015.5.501
[20]

Mihai Bostan, Gawtum Namah. Time periodic viscosity solutions of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2007, 6 (2) : 389-410. doi: 10.3934/cpaa.2007.6.389

 Impact Factor: 

Tools

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences