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Continuous dependence estimates for viscosity solutions of integro-PDEs
Authors:Espen R. Jakobsen  Kenneth H. Karlsen
Affiliation:a Department of Mathematical Sciences, Norwegian University of Science and Technology, N-7491 Trondheim, Norway
b Department of Mathematics, University of Bergen, Johs Brunsgt 12, N-5008 Bergen, Norway
c Centre of Mathematics for Applications (CMA), Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, N-0316 Oslo, Norway
Abstract:We present a general framework for deriving continuous dependence estimates for, possibly polynomially growing, viscosity solutions of fully nonlinear degenerate parabolic integro-PDEs. We use this framework to provide explicit estimates for the continuous dependence on the coefficients and the “Lévy measure” in the Bellman/Isaacs integro-PDEs arising in stochastic control/differential games. Moreover, these explicit estimates are used to prove regularity results and rates of convergence for some singular perturbation problems. Finally, we illustrate our results on some integro-PDEs arising when attempting to price European/American options in an incomplete stock market driven by a geometric Lévy process. Many of the results obtained herein are new even in the convex case where stochastic control theory provides an alternative to our pure PDE methods.
Keywords:Nonlinear degenerate parabolic integro-partial differential equation   Bellman equation   Isaacs equation   Viscosity solution   Continuous dependence estimate   Regularity   Vanishing viscosity method   Convergence rate
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