On the weak convergence of stochastic integrals on Skorokhod space: From general theory to applications within the realm of continuous-time random walks

Thesis

This thesis presents a comprehensive framework for the weak convergence of stochastic Itô integrals on Skorokhod space, under Skorokhod’s J1 and M1 topologies and based on a simple and tractable notion of good decompositions for integrators. We refine the classical theory, reveal new insights into convergence behaviours even in the classical J1 setting and address the sharpness of various key statements. We develop useful sufficient conditions for convergence and demonstrate the failure of general weak continuity results for strictly M1 convergent integrators. Furthermore, we identify a notable structural property, namely that the M1 tightness of local martingales generally implies their J1 tightness, thereby contributing to a deeper understanding of the nature of these stochastic processes. In addition, we study tightness results for stochastic integrals more broadly, under reduced regularity, and, in this regard, provide an extension to the M1 case for this very type of results existing within the J1 theory.

Moreover, we establish weak functional limit theorems for stochastic integrals driven by continuous-time random walks (CTRWs) and moving averages, both for the J1 and M1 topology. Building on this, we present weak approximation results for stochastic differential equations driven by time-changed Lévy processes, which are crucial for modeling anomalous diffusion across various fields. For stochastic integrals driven by strictly M1 convergent CTRWs and moving averages, where weak convergence may fail markedly, we determine natural classes of integrand processes that restore weak continuity. Not least, restricting to these classes yields functional limit theorems for certain stochastic delay differential equations driven by moving averages.

The theoretical advancements are supported by applications in econometrics, statistical mechanics, mathematical finance, and insurance mathematics, demonstrating the practical relevance of our results.

Authors: Wunderlich, F

• DOI: 10.5287/ora-koyo80yad
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: J1 topology; tightness; continuous-time random walks; limit theorems; stochastic integrals; Skorokhod; M1 topology; weak convergence; moving averages
• Subjects: Stochastic processes; Limit theorems (Probability theory); Stochastic analysis; Stochastic integrals; Random walks (Mathematics); Stochastic differential equations; Probability measures