Functional weak convergence of the financial gains for tick-by-tick models

Sojmark Andreas, London School of Economics
Wunderlich Fabrice, University of Oxford

Continuous time financial models driven by Levy processes should really be understood as scaling limits of some underlying tick-by-tick dynamics. Recently, Jacod and Ait-Sahalia has presented a thorough analysis of this, relying on Skorokhod’s M1 topology. Moreover, there is a fast growing literature on tick-by-tick models given by so-called continuous-time random walks and their convergence to stable subordinated Levy process in Skorokhod’s J1 or M1 topologies. A natural next question, then, is what can be said about the stability of the corresponding financial gains, across general classes of adapted trading strategies, as the tick-by-tick models converge to their scaling limit. The goal of this talk is to answer this question in as much generality as we can, leading to both positive and negative results for the J1 and M1 topologies. For concreteness, we will pay particular attention to models based on continuous-time random walks.

Area: CS55 - New probabilistic approaches in mathematical finance (Lorenzo Torricelli)

Keywords: Financial gains, tick-by-tick models, functional limit theorems, Skorokhod space, J1 and M1 topology

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