Pathwise integration and change of variable formulas for continuous paths with arbitrary regularity

Rama Cont,Nicolas Perkowski

doi:10.1090/btran/34

Abstract

We construct a pathwise integration theory, associated with a change of variable formula, for smooth functionals of continuous paths with arbitrary regularity defined in terms of the notion of p p th variation along a sequence of time partitions. For paths with finite p p th variation along a sequence of time partitions, we derive a change of variable formula for p p times continuously differentiable functions and show pointwise convergence of appropriately defined compensated Riemann sums. Results for functions are extended to regular path-dependent functionals using the concept of vertical derivative of a functional. We show that the pathwise integral satisfies an “isometry” formula in terms of p p th order variation and obtain a “signal plus noise” decomposition for regular functionals of paths with strictly increasing p p th variation. For less regular ( C p − 1 C^{p-1} ) functions we obtain a Tanaka-type change of variable formula using an appropriately defined notion of local time. These results extend to multidimensional paths and yield a natural higher-order extension of the concept of “reduced rough path”. We show that, while our integral coincides with a rough path integral for a certain rough path, its construction is canonical and does not involve the specification of any rough-path superstructure.

Highlights

In his seminal paper Calcul d’Ito sans probabilites [14], Hans Follmer provided a pathwise proof of the Itoformula, using the concept of quadratic variation along a sequence of partitions, defined as follows
[S] : [0, T ] → R+ defined by [S](t) = μ([0, t]) is called the quadratic variation of S along π. Extending this definition to vector-valued paths Follmer [14] showed that, for integrands of the form ∇f (S(t)) with f ∈ C2(Rd), one may define a Received by the editors April 18, 2018, and, in revised form, September 28, 2018. 2010 Mathematics Subject Classification
We show that Follmer’s pathwise Ito calculus may be extended to paths with arbitrary regularity, in a strictly pathwise setting, using the concept of pth variation along a sequence of time partitions

Summary

Introduction

In his seminal paper Calcul d’Ito sans probabilites [14], Hans Follmer provided a pathwise proof of the Itoformula, using the concept of quadratic variation along a sequence of partitions, defined as follows. For paths with finite pth variation along a sequence of time partitions, we derive a change of variable formula for p times continuously differentiable functions and show pointwise convergence of appropriately defined compensated Riemann sums. This result may be seen as the natural extension of the results of Follmer [14] to paths of lower regularity.

Pathwise calculus for paths with finite pth variation

Isometry relation and rough-smooth decomposition

Local times and higher-order Wuermli formula

Extension to multidimensional paths