Approximation of stochastic integrals with jumps via weighted BMO approach
Abstract
This article investigates discrete-time approximations of stochastic integrals driven by semimartingales with jumps via weighted bounded mean oscillation (BMO) approach. This approach enables Lp-estimates, p ∈ (2, ∞), for the approximation error depending on the weight, and it allows a change of the underlying measure which leaves the error estimates unchanged. To take advantage of this approach, we propose a new approximation scheme obtained from a correction for the Riemann approximation based on tracking jumps of the underlying semimartingale. We also discuss a way to optimize the approximation rate by adapting the discretization times to the setting. When the small jump activity of the semimartingale behaves like an α-stable process with α ∈ (1, 2), our scheme achieves under a regular regime the same convergence rate for the error as in Rosenbaum and Tankov [Ann. Appl. Probab. 24 (2014) 1002--1048]. Moreover, our approach extends to the case α ∈ (0, 1] and to the Lp-setting which are not treated there. As an application, we apply the methods in the special case where the semimartingale is an exponential L\'evy process to mean-variance hedging of European type options.