Local times of deterministic paths and model-free cadlag price paths and their application to mathematical finance

Abstract

This research explores the concept of local times for deterministic and c`adl`ag model-free paths, building on Hans F¨ollmer’s pioneering pathwise approach to Itˆo calculus. By utilising the notion of truncated variation, we establish the weak convergence of normalised interval crossing numbers and leverage this framework to construct illustrative examples on local times. Specifically, we construct a path based on the complement of the Cantor set that exhibits quadratic variation but lacks local time, along with a modified version that possesses local time. In addition, we develop a theorem on local times for functions of finite total variation, defining them as limits of interval crossings. This work introduces new change-of-variable formulas, extending the results of Bertoin and Yor. In their approach, Bertoin and Yor defined two notions of occupation measures for functions of finite variation based on level crossings and proved their continuity with respect to the Lebesgue measure using a Tanaka-Meyer-like change-of-variable formula. Following a similar spirit, we introduce two notions of local times based on interval crossings. However, our approach is more uniform and yields more general results. For instance, while Proposition 2 in Bertoin and Yor’s paper applies to continuously differentiable functions, we establish a similar result for locally Lipschitz functions. Lastly, we analyse the Peano curve, deriving conditions for the existence of its local times and emphasising the relationship between path structure and the existence of local times. This study advances the understanding of local times in a model-free setting, offering novel contributions to pathwise stochastic calculus.