Stability results for Hamilton-Jacobi equations with integro-differential terms and discontinuous Hamiltonians

Abstract

In this paper we prove a stability result for Hamilton-Jacobi equations with an integro-differential term for discontinuous Hamiltonians. This type of equations arises in various problems concerning, for example, the control of diffusion processes with jumps, the theory of large deviations for processes with jumps, and the theory of piecewise deterministic processes. 1. Introduction. The theory of viscosity solutions of Hamilton-Jacobi equations, introduced by M. G. Crandall and P. L. Lions [5], has given rise to numerous developments in such a way that from now on it covers all necessary tools for an effective use in applied mathematics. The case of Hamilton-Jacobi equations with continuous Hamiltonians was simplified by M. G. Crandall, L. C. Evans, and P. L. Lions [6] and thereafter extended, by P. L. Lions [8], to second order equations. The extension, of the notion of viscosity solutions to noncontinuous first order Hamiltonians, was first obtained by H. Ishii [4], and later by G. Barles and B. Perthame [2]. Short time after, A. Sayah [9] proved existence and uniqueness of viscosity solutions for the continuous Hamilton-Jacobi equations with an integro-differential term.In this paper we intend to prove a stability result for Hamilton-Jacobi equations with an integro-differential term for discontinuous Hamiltonians H defined by:

Integro-partial differential equations with singular terminal condition

“…Again in the non singular case, if the terminal function g is of linear growth, the relation between the FBSDE and the IPDE is obtained in [5]. Moreover several papers have studied the existence and the uniqueness of the solution of such IPDE (see among others [1], [8], [9] or [23]).…”

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.…”

On the Dirichlet problem for second-order elliptic integro-differential equations

“…By applying the Perron's Method (cf. [9,14] for local equations and [4,19,1,5,13] for non-local ones) together with the comparison result of [4] provides the existence of a solution u for this integro-differential equation such that |u| ≤ M .…”

“…We refer the reader to Ishii [14] or the User's guide [9] for the presentation of Perron's method which extends to the case of non-local equations because of the general stability result for integrodifferential equations of Bensaoud and Sayah [5] (see also [4]) and to [16,15,2,4] and references therein for comparison results for second order elliptic integro-differential equations.…”

“…Such operators enter in the general framework of (3) and the existence of µ x is obtained through the representation of integral (5). In order that the operator is well-defined, one assumes (6) |j(x, z)| ≤c|z| for somec > 0 and for any x, z ∈ R N .…”