Continuous dependence estimates for viscosity solutions of integro-PDEs

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.

Viscosity Solutions for a System of Integro-PDEs and Connections to Optimal Switching and Control of Jump-Diffusion Processes

“…Once we have the first part, then for the supersolution it says that at the point (t 0 , x 0 , y 0 ) we can ignore the termû i0 − M i0û , and then the second part follows as a consequence of Theorem 2.2 of [30].…”

“…On the other hand, the viscosity solution approach to nonlocal equations is still under development and is currently an active research area, cf. for example [1,2,6,7,17,29,30,31,38,39]. Contrary to its pure PDE counterpart, the available literature applying viscosity solutions to systems of integro-PDEs is very limited, but see [5] (switching systems are not covered).…”

“…However, due to the singular non-local part of these equations, it is not straightforward to give, as in the local case, a definition in terms of sub-or superjets. In this paper these jets are introduced via a "nonlocal" maximum principle of semi-continuous functions [30], see Lemma 4.1. We also refer to [7] for slightly different but (in this setting) equivalent way of doing this.…”

Fractal First-Order Partial Differential Equations

“…See also [25,23,1] and more recently [6][7][8]. A general theory for non-linear integro-partial differential equations is developed by Jakobsen and Karlsen [19,20]. Some of the ideas of [15] are adapted in [18] to prove that (2) has regular solutions if λ > 1.…”

Uniqueness for Integro-PDE in Hilbert Spaces

“…Papers and books [18,32,33,34,35,36,43,44,47,48,52,53,54,55,56,57,58,63,69] present various PDE, analytic and probabilistic approaches and connections with stochastic optimal control. The theory of viscosity solutions was introduced in [68,71,72] and its general theory was further developed in [2,3,4,11,12,15,16,17,19,23,24,25,37,40,41,70]. In particular, papers [24,70] contain higher order regularity results for viscosity solutions.…”