arXiv subjects
Publications and source records attributed to Andrzej Swiech.
We prove new comparison principles for viscosity solutions of non-linear integro-differential equations. The operators to which the method applies include but are not limited to those of Lévy-Itô type. The main idea is to use an optimal transport map to couple two different Lévy measures, and use the resulting coupling in a doubling of variables argument
The weak Harnack inequality for $L^p$-viscosity supersolutions of fully nonlinear second-order uniformly parabolic partial differential equations with unbounded coefficients and inhomogeneous terms is proved. It is shown that Hölder continuity of $L^p$-viscosity solutions is derived from the weak Harnack inequality for $L^p$-viscosity supersolutions. The local maximum principle for $L^p$-viscosity subsolutions and the Harnack inequality for $L^p$-viscosity solutions are also obtained. Several further remarks are presented when equations have superlinear growth in the first space derivatives.
We prove a stochastic representation formula for the viscosity solution of Dirichlet terminal-boundary value problem for a degenerate Hamilton-Jacobi-Bellman integro-partial differential equation in a bounded domain. We show that the unique viscosity solution is the value function of the associated stochastic optimal control problem. We also obtain the dynamic programming principle for the associated stochastic optimal control problem in a bounded domain.
The goal of this paper is to supplement the large deviation principle of the Freidlin--Wentzell theory on exit problems for diffusion processes with results of classical central limit theorem kind. We describe a class of situations where conditioning on exit through unlikely locations leads to a Gaussian scaling limit for the exit distribution. Our results are based on Doob's h-transform and new asymptotic convergence gradient estimates for elliptic nonlinear equations that allow to reduce the problem to the Levinson case. We devote a separate section to a rigorous and general discussion of h-transform.
We prove a large deviation principle result for solutions of abstract stochastic evolution equations perturbed by small Levy noise. We use general large deviations theorems of Varadhan and Bryc, viscosity solutions of integro-partial differential equations in Hilbert spaces, and deterministic optimal control methods. The Laplace limit is identified as a viscosity solution of a Hamilton-Jacobi-Bellman equation of an associated control problem. We also establish exponential moment estimates for solutions of stochastic evolution equations driven by Levy noise. General results are applied to stochastic hyperbolic equations perturbed by subordinated Wiener process.