Existence, uniqueness and regularity of solutions to systems of nonlocal obstacle problems related to optimal switching

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:475

Article

Lundstrom, Niklas L. P.¹  Olofsson, Marcus²  Onskog, Thomas³ 

[1] Umea Univ, Dept Math & Math Stat, SE-90187 Umea, Sweden

[2] Uppsala Univ, Dept Math, SE-75106 Uppsala, Sweden

[3] KTH Royal Inst Technol, Dept Math, SE-10044 Stockholm, Sweden

关键词: Variational inequality;    Nonlocal operator;    Partial integro-differential operator;    Optimal switching problem;    Viscosity solution;    Regularity;   

DOI  :  10.1016/j.jmaa.2018.11.003

来源: Elsevier

【 摘 要 】

We study viscosity solutions to a system of nonlinear degenerate parabolic partial integro-differential equations with interconnected obstacles. This type of problem occurs in the context of optimal switching problems when the dynamics of the underlying state variable is described by an n-dimensional Levy process. We first establish a continuous dependence estimate for viscosity sub- and supersolutions to the system under mild regularity, growth and structural assumptions on the partial integro-differential operator and on the obstacles and terminal conditions. Using the continuous dependence estimate, we obtain the comparison principle and uniqueness of viscosity solutions as well as Lipschitz regularity in the spatial variables. Our main contribution is construction of suitable families of viscosity sub- and supersolutions which we use as barrier functions to prove Holder continuity in the time variable, and, through Perron's method, existence of a unique viscosity solution. This paper generalizes parts of the results of Biswas, Jakobsen and Karlsen (2010) [5] and of Lundstrom, Nystrom and Olofsson (2014) [21,22] to hold for more general systems of equations. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free