Error estimates for approximate solutions to Bellman equations associated with controlled jump-diffusions

Abstract

Abstract. We derive error estimates for approximate (viscosity) solutions of Bellman equations associated to controlled jump-diffusion processes, which are fully nonlinear integro-partial differential equations. Two main results are obtained: (i) error bounds for a class of monotone approximation schemes, which under some assumptions includes finite difference schemes, and (ii) bounds on the error induced when the original Lévy measure is replaced by a finite measure with compact support, an approximation process that is commonly used when designing numerial schemes for integro-partial differential equations. Our proofs use and extend techniques introduced by Krylov and Barles-Jakobsen.

Numerical methods for controlled Hamilton-Jacobi-Bellman PDEs in finance

“…In some cases, it may also be a nontrivial problem to solve the local control problem (6.12). This may be especially difficult if jump processes are modelled, which results in a controlled partial integrodifferential equation (PIDE) [24]. In addition, the policy iteration scheme does not guarantee global convergence of (6.7) for HJBI equations.…”

Error estimates for finite difference-quadrature schemes for a class of nonlocal Bellman equations with variable diffusion

“…The problem of error estimates for numerical schemes for fully nonlinear degenerate integro-PDEs is mostly an untouched area, except for the recent treatment in [19]. In [19], adapting the methods of Krylov and Barles-Jakobsen, the authors derive error estimates for a class of approximation schemes for (1.1).…”

“…In [19], adapting the methods of Krylov and Barles-Jakobsen, the authors derive error estimates for a class of approximation schemes for (1.1). However, to apply the results from [19] to a particular numerical scheme certain nontrivial conditions have to be satisfied, conditions which could be heard to verify in general.…”

“…In [19], adapting the methods of Krylov and Barles-Jakobsen, the authors derive error estimates for a class of approximation schemes for (1.1). However, to apply the results from [19] to a particular numerical scheme certain nontrivial conditions have to be satisfied, conditions which could be heard to verify in general. As a working example the authors consider an upwind finite difference-quadrature scheme for the Bellman equation where the diffusion matrix a θ is independent of the spatial variable x for each admissible control θ.…”

“…As a working example the authors consider an upwind finite difference-quadrature scheme for the Bellman equation where the diffusion matrix a θ is independent of the spatial variable x for each admissible control θ. It is not clear, from [19], what happens if the diffusion matrix depends on the spatial variable.…”