2017/11/13 by Niklas L. P. Lundström, Lundström, Niklas L. P., Marcus Olofsson +3
Engineering · Mathematics · #35R09 #45G15 #49L25 #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1711.04665
openalex publication_date 2017/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study viscosity solutions to a system of nonlinear degenerate parabolic\npartial integro-differential equations with interconnected obstacles. This type\nof problem occurs in the context of optimal switching problems when the\ndynamics of the underlying state variable is described by an n-dimensional\nLevy process. We first establish a continuous dependence estimate for viscosity\nsub- and supersolutions to the system under mild regularity, growth and\nstructural assumptions on the partial integro-differential operator and on the\nobstacles and terminal conditions. Using the continuous dependence estimate, we\nobtain the comparison principle and uniqueness of viscosity solutions as well\nas Lipschitz regularity in the spatial variables. Our main contribution is\nconstruction of suitable families of viscosity sub- and supersolutions which we\nuse as barrier functions to prove H "older continuity in the time variable,\nand, through Perron's method, existence of a unique viscosity solution. This\npaper generalizes parts of the results of Biswas, Jakobsen and Karlsen (2010)\nand of Lundstr "om, Nystr "om and Olofsson (2014) to hold for more general\nsystems of equations.\n