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

## 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 variation along a sequence of time partitions. For paths with finite th variation along a sequence of time partitions, we derive a change of variable formula for th 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 order variation and obtain a “signal plus noise” decomposition for regular functionals of paths with strictly increasing th variation. For less regular th( 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.

## Introduction

In his seminal paper *Calcul d’Itô sans probabilités* Reference 14, Hans Föllmer provided a pathwise proof of the Itô formula, using the concept of *quadratic variation along a sequence of partitions*, defined as follows. A path is said to have finite quadratic variation along the sequence of partitions if for any the sequence of measures ,

converges weakly to a measure without atoms. The continuous increasing function defined by is then called the quadratic variation of along Extending this definition to vector-valued paths Föllmer .Reference 14 showed that, for integrands of the form with one may define a pathwise integral , as a pointwise limit of Riemann sums along the sequence of partitions and he obtained an Itô (change of variable) formula for in terms of this pathwise integral: for ,

where

This result has many interesting ramifications and applications in the pathwise approach to stochastic analysis, and has been extended in different ways, to less regular functions using the notion of pathwise local time Reference 2Reference 10Reference 24, as well as to path-dependent functionals and integrands Reference 1Reference 7Reference 8Reference 25.

The central role played by the concept of quadratic variation has led to the presumption that they do not extend to less regular paths with infinite quadratic variation. Integration theory and change of variables formulas for processes with infinite quadratic variation, such as fractional Brownian motion and other fractional processes, have relied on probabilistic, rather than pathwise constructions Reference 5Reference 9Reference 18. Furthermore, the change of variable formulae obtained using these methods are valid for a restricted range of Hurst exponents (see Reference 23 for an overview).

In this work, we show that Föllmer’s pathwise Itô calculus may be extended to paths with arbitrary regularity, in a strictly pathwise setting, using the concept of variation along a sequence of time partitions. For paths with finite th variation along a sequence of time partitions, we derive a change of variable formula for th 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 Föllmer Reference 14 to paths of lower regularity. Our results apply in particular to paths of fractional Brownian motions with arbitrary Hurst exponent, and yield pathwise proofs for results previously derived using probabilistic methods, without any restrictions on the Hurst exponent.

Using the concept of the vertical derivative of a functional Reference 8, we extend these results to regular path-dependent functionals of such paths. We obtain an “isometry” formula in terms of order variations for the pathwise integral and a “signal plus noise” decomposition for regular functionals of paths with strictly increasing th variation, extending the results of thReference 1 obtained for the case to arbitrary even integers .

The extension to less regular (i.e., not times differentiable) functions is more delicate and requires defining an appropriate higher-order analogue of semimartingale local time, which we introduce through an appropriate spatial localization of the order variation. Using this higher-order concept of local time, we obtain a Tanaka-type change of variable formula for less regular (i.e., th times differentiable) functions. We conjecture that these results apply in particular to paths of fractional Brownian motion and other fractional processes.

Finally, we consider extensions of these results to multidimensional paths and link them with rough path theory; the corresponding concepts yield a natural higher-order extension to the concept of “reduced rough path” introduced by Friz and Hairer Reference 17, Chapter 5.

### Outline

Section 1 introduces the notion of variation along a sequence of partitions and derives a change of variable formula for th times continuously differentiable functions of paths with finite variation (Theorem th1.5). An extension of these results to path-dependent functionals is discussed in Section 1.3: Theorem 1.10 gives a functional change of variable formula for regular functionals of paths with finite variation. th

Section 2 studies the corresponding pathwise integral in more detail. We first show (Theorem 2.1) that the integral exhibits an “isometry” property in terms of the order variation and use this property to obtain a unique “signal plus noise” decomposition where the components are discriminated in terms of their th order variation (Theorem th2.3).

The extension of these concepts to multidimensional paths and the relation to the concept of “reduced rough paths” are discussed in Section 4.

## 1. Pathwise calculus for paths with finite variation th

### 1.1. variation along a sequence of partitions th

We introduce, in the spirit of Föllmer Reference 14, the concept of variation along a sequence of partitions th with Define the .*oscillation* of along as

Here and in the following we write to indicate that and are both in and are immediate successors (i.e., and ).

The following lemma gives a simple characterization of this property.

Indeed, the weak convergence of measures on is equivalent to the pointwise convergence of their cumulative distribution functions at all continuity points of the limiting cumulative distribution function, and if the limiting cumulative distribution function is continuous, the convergence is uniform.

### 1.2. Pathwise integral and change of variable formula

A key observation of Föllmer Reference 14 was that, for Definition ,1.1 is sufficient to obtain a pathwise Itô formula for ( functions of ) We will show that in fact Föllmer’s argument may be applied for any even integer . .