A posteriori error analysis for a class of integral equations and variational inequalities

Abstract

We consider elliptic and parabolic variational equations and inequalities governed by integro-differential operators of order 2s ∈ (0, 2]. Our main motivation is the pricing of European or American options under Lévy processes, in particular pure jump processes or jump diffusion processes with tempered stable processes. The problem is discretized using piecewise linear finite elements in space and the implicit Euler method in time. We construct a residual-type a posteriori error estimator which gives a computable upper bound for the actual error in H s -norm. The estimator is localized in the sense that the residuals are restricted to the discrete non-contact region. Numerical experiments illustrate the accuracy of the space and time estimators, and show that they can be used to measure local errors and drive adaptive algorithms.

Inexact arithmetic considerations for direct control and penalty methods: American options under jump diffusion

“…This contract has been used as severe test case by several authors [1,26,24]. In the no-jump case, the exercise region is not simply connected to the boundary, hence the direct method in [5] cannot be used (at least in straightforward fashion) and an iterative method is required.…”

Methods for Pricing American Options under Regime Switching

“…This contract has been used as a severe test case by several authors [1,41,32]. The variable timestep selector described in [16] is used.…”

A Remark on the A Posteriori Error Analysis of Discontinuous Galerkin Methods for the Obstacle Problem

“…In that case the error estimator takes a simpler form which does not involve min/max functions. For instance we can see this by comparing the results in [33,Section 3] with the results in [33,Section 4], also the results in [5,Section 1.3] with the results in [5,Section 1.4] and by noting the results in [7,8,30]. But, in general, the error analysis for general obstacle leads to several difficulties and the error estimator takes a complicated form [5,25,33] involving min/max functions.…”

“…A posteriori error analysis for the obstacle problem has received attention in the last years [5,7,8,17,30,33]. In the a posteriori error analysis of the obstacle problem, it is often found convenient to assume that the underlying obstacle is a global affine function or a function from the finite element space [7,8,30]. In that case the error estimator takes a simpler form which does not involve min/max functions.…”

“…We refer to [11,12,22,23,36,37] for the a priori error analysis of finite element methods for the obstacle problem. A posteriori error analysis for the obstacle problem has received attention in the last years [5,7,8,17,30,33]. In the a posteriori error analysis of the obstacle problem, it is often found convenient to assume that the underlying obstacle is a global affine function or a function from the finite element space [7,8,30].…”