On the strong maximum principle for second order nonlinear parabolic integro-differential equations

This paper is concerned with the study of the Strong Maximum Principle for semicontinuous viscosity solutions of fully nonlinear, second-order parabolic integro-differential equations. We study separately the propagation of maxima in the horizontal component of the domain and the local vertical propagation in simply connected sets of the domain. We give two types of results for horizontal propagation of maxima: one is the natural extension of the classical results of local propagation of maxima and the other comes from the structure of the nonlocal operator. As an application, we use the Strong Maximum Principle to prove a Strong Comparison Result of viscosity sub and supersolution for integro-differential equations.

[1]  Hitoshi Ishii,et al.  The maximum principle for semicontinuous functions , 1990, Differential and Integral Equations.

[2]  Martino Bardi,et al.  Propagation of Maxima and Strong Maximum Principle for Viscosity Solutions of Degenerate Elliptic Equations , 2000 .

[3]  E. Jakobsen,et al.  CONTINUOUS DEPENDENCE ESTIMATES FOR VISCOSITY SOLUTIONS OF INTEGRO-PDES , 2005 .

[4]  Anna Lisa Amadori,et al.  Nonlinear integro-differential evolution problems arising in option pricing: a viscosity solutions approach , 2003, Differential and Integral Equations.

[5]  G. Barles,et al.  Second-order elliptic integro-differential equations: viscosity solutions' theory revisited , 2007, math/0702263.

[6]  G. Barles,et al.  H\^older continuity of solutions of second-order non-linear elliptic integro-differential equations , 2010, 1009.0685.

[7]  Kenneth H. Karlsen,et al.  Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: A viscosity solution approach , 2001, Finance Stochastics.

[8]  Francesca Da Lio,et al.  REMARKS ON THE STRONG MAXIMUM PRINCIPLE FOR VISCOSITY SOLUTIONS TO FULLY NONLINEAR PARABOLIC EQUATIONS , 2004 .

[9]  Martino Bardi,et al.  Propagation of maxima and Strong Maximum Principle for viscosity solutions of degenerate elliptic equations. II: Concave operators , 2003 .

[10]  H. Soner.  Optimal control with state-space constraint I , 1986 .

[11]  W. Woyczynski.  Lévy Processes in the Physical Sciences , 2001 .

[12]  M. Arisawa,et al.  A new definition of viscosity solutions for a class of second-order degenerate elliptic integro-differential equations , 2006 .

[13]  Huy En Pham.  Optimal Stopping of Controlled Jump Diiusion Processes: a Viscosity Solution Approach , 1998 .

[14]  P. Lions,et al.  Viscosity solutions of fully nonlinear second-order elliptic partial differential equations , 1990 .

[15]  G. Barles,et al.  Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations , 2011 .

[16]  L. Nirenberg.  A strong maximum principle for parabolic equations , 1953 .

[17]  J. Coville.  Remarks on the strong maximum principle for nonlocal operators , 2008 .

[18]  Sayah Awatif.  Equqtions D'Hamilton-Jacobi Du Premier Ordre Avec Termes Intégro-Différentiels: Partie II: Unicité Des Solutions De Viscosité , 1991 .

[19]  E. Jakobsen,et al.  A “maximum principle for semicontinuous functions” applicable to integro-partial differential equations , 2006 .

[20]  Sayah Awatif,et al.  Equqtions D'Hamilton-Jacobi Du Premier Ordre Avec Termes Intégro-Différentiels , 2007 .

[21]  H. Ishii.  On uniqueness and existence of viscosity solutions of fully nonlinear second‐order elliptic PDE's , 1989 .

[22]  G. Barles.  Solutions de viscosité des équations de Hamilton-Jacobi , 1994 .

[23]  R. Jensen.  The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations , 1988 .

[24]  C. Imbert.  A non-local regularization of first order Hamilton–Jacobi equations , 2005 .

[25]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.