Weighted BMO, Riemann-Liouville Type Operators, and Approximation of Stochastic Integrals in Models with Jumps

Abstract

This thesis investigates the interplay between weighted bounded mean oscillation (BMO), Riemann–Liouville type operators applied to càdlàg processes, real interpolation, gradient type estimates for functionals on the Lévy–Itô space, and approximation for stochastic integrals with jumps. There are two main parts included in this thesis. The first part discusses the connections between the approximation problem in L2 or in weighted BMO, Riemann–Liouville type operators, and the real interpolation theory in a general framework (Chapter 3). The second part provides various applications of results in the first part to several models: diffusions in the Brownian setting (Section 3.5) and certain jump models (Chapter 4) for which the (exponential) Lévy settings are typical examples (Chapter 6 and Chapter 7). Especially, for the models with jumps we propose a new approximation scheme based on an adjustment of the Riemann approximation of stochastic integrals so that one can effectively exploit the features of weighted BMO. In our context, making a bridge from the first to the second part requires gradient type estimates for a semigroup acting on Hölder functions in both the Brownian setting (Section 3.5) and the (exponential) Lévy setting (Chapter 5). In the latter case, we consider a kind of gradient processes appearing naturally from the Malliavin derivative of functionals of the Lévy process, and we show how the gradient behaves in time depending on the “direction” one tests.