# 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:

show all references

##### References:
 [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: