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 $p$th variation along a sequence of time partitions. For paths with finite $p$th 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.

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 $p$th order variation and obtain a “signal plus noise” decomposition for regular functionals of paths with strictly increasing $p$th variation. For less regular ($C^{p-1}$) 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.

Introduction

In his seminal paper Calcul d’Itô sans probabilités 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 $S\in C([0,T],\mathbb{R})$ is said to have finite quadratic variation along the sequence of partitions $\pi _n=(0=t^{n}_0<t^n_1<\cdots <t^{n}_{N(\pi _n)}=T)$ if for any $t\in [0,T]$, the sequence of measures

$$\begin{equation*} \mu ^n :=\sum _{[t^n_j, t^n_{j + 1}] \in \pi _n} \delta (\cdot - t_j) |S(t^n_{j + 1}) - S(t^n_j)|^2 \end{equation*}$$

converges weakly to a measure $\mu$ without atoms. The continuous increasing function $[S] \colon [0,T]\to \mathbb{R}_+$ defined by $[S](t)=\mu ([0,t])$ is then called the quadratic variation of $S$ along $\pi$. Extending this definition to vector-valued paths Föllmer 14 showed that, for integrands of the form $\nabla f( S(t))$ with $f\in C^2(\mathbb{R}^d)$, one may define a pathwise integral $\int \nabla f(S(t))dS$ as a pointwise limit of Riemann sums along the sequence of partitions $(\pi _n)$ and he obtained an Itô (change of variable) formula for $f( S(t))$ in terms of this pathwise integral: for $f\in C^2(\mathbb{R}^d), t\in [0,T]$,

$$\begin{equation*} f(S(t))= \int _0^t \langle \nabla f(S(s)), \mathrm{d}S(s)\rangle + \frac{1}{2}\int _0^t \langle \nabla ^2 f(S(s)), \mathrm{d}[S](s)\rangle , \end{equation*}$$

where

$$\begin{equation*} \int _0^t \langle \nabla f(S(s)) , \mathrm{d}S(s) \rangle :=\mathop{\lim }_{n\to \infty }\sum _{[t^n_j, t^n_{j + 1}] \in \pi _n} \langle \nabla f(S(t)), ( S(t^n_{j + 1} \wedge t) - S(t^n_j \wedge t))\rangle . \end{equation*}$$

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 21024, as well as to path-dependent functionals and integrands 17825.

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 5918. Furthermore, the change of variable formulae obtained using these methods are valid for a restricted range of Hurst exponents (see 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 $p$th variation along a sequence of time partitions. For paths with finite $p$th 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 Föllmer 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 8, we extend these results to regular path-dependent functionals of such paths. We obtain an “isometry” formula in terms of $p$th order variations for the pathwise integral and a “signal plus noise” decomposition for regular functionals of paths with strictly increasing $p$th variation, extending the results of 1 obtained for the case $p=2$ to arbitrary even integers $p\geq 2$.

The extension to less regular (i.e., not $p$ 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 $p$th order variation. Using this higher-order concept of local time, we obtain a Tanaka-type change of variable formula for less regular (i.e., $p-1$ 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 17, Chapter 5.

Outline

Section 1 introduces the notion of $p$th variation along a sequence of partitions and derives a change of variable formula for $p$ times continuously differentiable functions of paths with finite $p$th variation (Theorem 1.5). 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 $p$th variation.

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 $p$th order variation and use this property to obtain a unique “signal plus noise” decomposition where the components are discriminated in terms of their $p$th order variation (Theorem 2.3).

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 $p$th variation

1.1. $p$th variation along a sequence of partitions

We introduce, in the spirit of Föllmer 14, the concept of $p$th variation along a sequence of partitions $\pi _n=\{t_0^n, \dots , t^n_{N(\pi _n)}\}$ with $t_0^n=0<\ldots < t^n_k<\ldots < t^n_{N(\pi _n)}=T$. Define the oscillation of $S \in C([0,T],\mathbb{R})$ along $\pi _n$ as

$$\begin{equation*} \operatorname {osc}(S,\pi _n) := \max _{[t_j,t_{j+1}] \in \pi _n} \max _{r,s \in [t_j,t_{j+1}]} |S(s) - S(r)|. \end{equation*}$$

Here and in the following we write $[t_j,t_{j+1}] \in \pi _n$ to indicate that $t_j$ and $t_{j+1}$ are both in $\pi _n$ and are immediate successors (i.e., $t_j < t_{j+1}$ and $\pi _n \cap (t_j , t_{j+1}) = \emptyset$).

Definition 1.1 ($p$th variation along a sequence of partitions).

Let $p>0$. A continuous path $S \in C([0,T],\mathbb{R})$ is said to have a $p$th variation along a sequence of partitions $\pi =(\pi _n)_{n\geq 1}$ if $\operatorname {osc}(S,\pi _n)\to 0$ and the sequence of measures

$$\begin{equation*} \mu ^n :=\sum _{[t_j, t_{j + 1}] \in \pi _n} \delta (\cdot - t_j) |S(t_{j + 1}) - S(t_j)|^p \end{equation*}$$

converges weakly to a measure $\mu$ without atoms. In that case we write $S \in V_p(\pi )$ and $[S]^p(t) := \mu ([0,t])$ for $t \in [0,T]$, and we call $[S]^p$ the $p$th variation of $S$.

Remark 1.2.

(1)

Functions in $V_p(\pi )$ do not necessarily have finite $p$-variation in the usual sense. Recall that the $p$-variation of a function $f \in C([0,T],\mathbb{R})$ is defined as 11$$\begin{equation*} \|f \|_{\operatorname {p-var}} :=\Big (\sup _{\pi \in \Pi ([0,T])} \sum _{[t_j, t_{j + 1}] \in \pi } |f(t_{j + 1}) - f(t_j)|^p\Big )^{1/p}, \end{equation*}$$

where the supremum is taken over the set $\Pi ([0,T])$ of all partitions $\pi$ of $[0,T]$. A typical example is the Brownian motion $B$, which has quadratic variation $[B]^2(t) = t$ along any refining sequence of partitions almost surely while at the same time having infinite 2-variation almost surely 1129:$$\begin{equation*} \mathbb{P}\left( \|B\|_{\operatorname {2-var}} = \infty \right) =1. \end{equation*}$$

(2)

If $S \in V_p(\pi )$ and $q > p$, then $S \in V_q(\pi )$ with $[S]^q \equiv 0$.

The following lemma gives a simple characterization of this property.

Lemma 1.3.

Let $S \in C([0,T],\mathbb{R})$. $S \in V_p(\pi )$ if and only if there exists a continuous function $[S]^p$ such that

$$\begin{equation} \forall t\in [0,T],\qquad \sum _{\substack{[t_j, t_{j + 1}] \in \pi _n: \\ t_j \le t}} |S(t_{j + 1}) - S(t_j)|^p\mathop{\longrightarrow }^{n\to \infty } [S]^p(t).{\tag{1}} \end{equation}$$

If this property holds, then the convergence in 1 is uniform.

Indeed, the weak convergence of measures on $[0,T]$ 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.

Example 1.4.

If $B$ is a fractional Brownian motion with Hurst index $H \in (0,1)$ and $\pi _n = \{kT/n: k \in \mathbb{N}_0\} \cap [0,T]$, then $B \in V_{1/H}(\pi )$ and $[B]^{1/H}(t) = t \mathbb{E}[|B_1|^{1/H}]$; see 2627.

1.2. Pathwise integral and change of variable formula

A key observation of Föllmer 14 was that, for $p=2$, Definition 1.1 is sufficient to obtain a pathwise Itô formula for ($C^2$) functions of $S \in V_2(\pi )$. We will show that in fact Föllmer’s argument may be applied for any even integer $p$.

Theorem 1.5 (Change of variable formula for paths with finite $p$th variation).

Let $p \in \mathbb{N}$ be even, let $(\pi _n)$ be a given sequence of partitions, and let $S \in V_p(\pi )$. Then for every $f \in C^p(\mathbb{R},\mathbb{R})$ the pathwise change of variable formula

$$\begin{equation*} f(S(t)) - f(S(0)) = \int _0^t f'(S(s))\mathrm{d}S(s) + \frac{1}{p!} \int _0^t f^{(p)}(S(s)) \mathrm{d}[S]^p(s) \end{equation*}$$

holds, where the integral

$$\begin{equation*} \int _0^t f'(S(s))\mathrm{d}S(s) :=\lim _{n \rightarrow \infty } \sum _{[t_j, t_{j + 1}] \in \pi _n} \sum _{k=1}^{p-1} \frac{f^{(k)} (S(t_j))}{k!} (S(t_{j + 1} \wedge t) - S(t_j \wedge t))^k \end{equation*}$$

is defined as a (pointwise) limit of compensated Riemann sums.

Proof.

Applying a Taylor expansion at order $p$ to the increments of $f(S)$ along the partition, we obtain

$$\begin{equation} \begin{aligned} f (S(t)) - f (S(0)) & = \sum _{[t_j, t_{j + 1}] \in \pi _n} (f (S(t_{j + 1}\wedge t)) - f (S(t_j \wedge t)))\\ & = \sum _{[t_j, t_{j + 1}] \in \pi _n} \sum _{k=1}^p \frac{f^{(k)} (S(t_j))}{k!} (S(t_{j + 1}\wedge t) - S(t_j \wedge t))^k\\ &\qquad + \sum _{[t_j, t_{j + 1}] \in \pi _n} \int _0^1 \mathrm{d}\lambda \frac{(1 - \lambda )^{p - 1}}{(p - 1) !} (S(t_{j + 1} \wedge t) - S(t_j \wedge t))^p \times \big (f^{(p)} (S(t_j) + \lambda (S(t_{j + 1} \wedge t) - S(t_j \wedge t))) - f^{(p)} (S(t_j))\big ) . \end{aligned} {\tag{2}} \end{equation}$$

Since the image of $(S(t))_{t \in [0,T]}$ is compact, we may assume without loss of generality that $f$ is compactly supported; then the remainder on the right hand side is bounded by

$$\begin{align*} \lefteqn{\Big |\sum _{[t_j, t_{j + 1}] \in \pi _n} \int _0^1 \mathrm{d}\lambda \frac{(1 - \lambda )^{p - 1}}{(p - 1) !} (S(t_{j + 1} \wedge t) - S(t_j \wedge t))^p}\qquad \\ &\qquad \qquad \qquad \times \big (f^{(p)} (S(t_j) + \lambda (S(t_{j + 1} \wedge t) - S(t_j \wedge t))) - f^{(p)}(S(t_j))\big ) \Big | \\ & \le C(f, S, \pi _n, p) \mu _n([0,t]) \end{align*}$$

with a constant $C(f, S, \pi _n, p) > 0$ that converges to zero for $n \rightarrow \infty$, and therefore the remainder vanishes for $n \to \infty$. Since $S \in V_p(\pi )$ we know that

$$\begin{equation*} \lim _{n\to \infty } \sum _{[t_j, t_{j + 1}] \in \pi _n} \frac{f^{(p)} (S(t_j))}{p!} (S(t_{j + 1}\wedge t) - S(t_j \wedge t))^p = \frac{1}{p!} \int _0^t f^{(p)}(S(s)) \mathrm{d}[S]^p(s), \end{equation*}$$

and therefore we obtain from 2

$$\begin{align*} \lefteqn{\lim _{n \to \infty } \sum _{[t_j, t_{j + 1}] \in \pi _n} \sum _{k=1}^{p-1} \frac{f^{(k)} (S(t_j))}{k!} (S(t_{j + 1} \wedge t) - S(t_j \wedge t))^k}\qquad \\ &= f(S(t)) - f(S(0)) - \frac{1}{p!} \int _0^t f^{(p)}(S(s)) \mathrm{d}[S]^p(s), \end{align*}$$

and we simply define $\int _0^t f'(S(s)) \mathrm{d}S(s)$ as the limit on the left hand side.

■

Remark 1.6 (Relation with Young integration and rough path integration).

The expression

$$\begin{equation*} \sum _{[t_j, t_{j + 1}] \in \pi _n} \sum _{k=1}^{p-1} \frac{f^{(k)} (S(t_j))}{k!} (S(t_{j} \wedge t) - S(t_j \wedge t))^k \end{equation*}$$

is a “compensated Riemann sum”. Note however that, given the assumptions on $S$, the pathwise integral appearing in the formula cannot be defined as a Young integral, even after substracting the compensating terms. This relates to the observation in Remark 1.2 that $p$-variation can be infinite for $S\in V_p(\pi )$.

When $p=2$ it reduces to an ordinary (left) Riemann sum. For $p>2$ such compensated Riemann sums appear in the construction of “rough path integrals” 1719. Let $X \in C^\alpha ([0,T],\mathbb{R})$ be $\alpha$-Hölder continuous for some $\alpha \in (0,1)$, and write $q = \lfloor \alpha ^{-1} \rfloor$. We can enhance $X$ uniquely into a (weakly) geometric rough path $(\mathbb{X}^1_{s,t}, \mathbb{X}^2_{s,t}, \dots , \mathbb{X}^q_{s,t})_{0 \le s \le t \le T}$, where $\mathbb{X}^k_{s,t} :=(X(t) - X(s))^k/k!$. Moreover, for $g \in C^{q+1}(\mathbb{R},\mathbb{R})$ the function $g'(X)$ is controlled by $X$ with Gubinelli derivatives

$$\begin{align*} g'(X(t)) - g'(X(s)) & = \sum _{k=1}^{q-1} \frac{g^{(k+1)}(X(s))}{k!} (X(t) - X(s))^k + O(|t-s|^{q \alpha }) \\ & = \sum _{k=1}^{q-1} g^{(k+1)}(X(s)) \mathbb{X}^k_{s,t} + O(|t-s|^{q \alpha }), \end{align*}$$

and therefore the controlled rough-path integral $\int _0^t g'(X(s)) \mathrm{d}X(s)$ is given by

$$\begin{equation*} \lim _{|\pi |\to 0} \sum _{[t_j, t_{j+1}] \in \pi } \sum _{k=1}^q g^{(k)}(X(s)) \mathbb{X}^k_{s,t} = \lim _{|\pi |\to 0} \sum _{[t_j, t_{j+1}] \in \pi } \sum _{k=1}^q g^{(k)}(X(s)) \frac{(X(t) - X(s))^k}{k!}, \end{equation*}$$

where $|\pi |$ denotes the mesh size of the partition $\pi$, and which is exactly the type of compensated Riemann sum that we used to define our integral. The link between our approach and rough-path integration is explained in more detail in Section 4.2 below.

Remark 1.7.

In principle we could apply similar arguments for odd integers $p$ if instead of $S \in V_p(\pi )$ we assumed that $\sum _{[t_j, t_{j + 1}] \in \pi _n} \delta (\cdot - t_j) (S(t_{j + 1}) - S(t_j))^p$ converges to a signed measure. However, for odd $p$ we typically expect the limit to be zero; see the appendix for a prototypical example. So to slightly simplify the presentation, we restrict our attention to even $p$.

Remark 1.8.

A notion similar to our definition of $p$th variation was introduced by Errami and Russo 13, in the (probabilistic and not pathwise) context of stochastic calculus via regularization 28. For $p=3$, Errami and Russo prove an Itô-type formula that is similar to the one in Theorem 1.5. However, since they use a definition of the integral $\int _0^t f'(S(s)) \mathrm{d}S(s)$ that does not take the higher-order compensation terms into account, their approach is limited to $p=3$. Gradinaru, Russo, and Vallois 18 extended this approach to $p=4$ for functions of a fractional Brownian motion with Hurst index $H\ge 1/4$, a result which relies heavily on the Gaussian properties of fractional Brownian motion.

The key ingredient of our approach is to define the integral using compensated Riemann sums which, compared with previous work, drastically simplifies the derivation of the change of variable formula for arbitrary (even) $p$ in a strictly pathwise setting without any use of probabilistic notions of convergence.

1.3. Extension to path-dependent functionals

An important generalization of Föllmer’s pathwise Itô formula is to the case of path-dependent functionals 8 of paths $S \in V_2(\pi )$ using Dupire’s functional derivative 12; see 7 for an overview. We extend here the functional change of variable formula of Cont and Fournié 8 to functionals of paths $S \in V_p(\pi )$, where $p$ is any even integer.

Let $D([0,T],\mathbb{R})$ be the space of càdlàg paths from $[0,T]$ to $\mathbb{R}$ and write

$$\begin{equation*} \omega _t(s) = \omega (s \wedge t) \end{equation*}$$

for the path $\omega$ stopped at time $t$. Let

$$\begin{equation*} \Lambda _T :=\{ (t, \omega _t): (t,\omega ) \in [0,T] \times D([0,T],\mathbb{R})\} \end{equation*}$$

be the space of stopped paths. This is a complete metric space equipped with

$$\begin{equation*} d_\infty ((t,\omega ), (t',\omega ')) := \sup _{s \in [0,T]} |\omega (s \wedge t) - \omega '(s \wedge t')| + |t-t'| = \|\omega _t - \omega _{t'}\|_\infty + |t - t'|. \end{equation*}$$

We will also need to stop paths “right before” a given time, and set for $t>0$

$$\begin{equation*} \omega _{t-}(s) :=\begin{cases} \omega (s), & s < t, \\ \mathop{\lim }_{r \uparrow t} \omega (r), & s \ge t, \end{cases} \end{equation*}$$

while $\omega _{0-} :=\omega _0$. We first recall some concepts from the non-anticipative functional calculus 78.

Definition 1.9.

A non-anticipative functional is a map $F\colon \Lambda _T \to \mathbb{R}$. Let $F$ be a non-anticipative functional.

i.

We write $F \in \mathbb{C}_l^{0,0}(\Lambda _T)$ if for all $t \in [0,T]$ the map $F(t,\cdot ) \colon D([0,T],\mathbb{R}) \to \mathbb{R}$ is continuous and if for all $(t,\omega ) \in \Lambda _T$ and all $\varepsilon > 0$ there exists $\delta > 0$ such that for all $(t',\omega ') \in \Lambda _T$ with $t'<t$ and $d_\infty ((t,\omega ),(t',\omega ')) < \delta$ we have $|F(t,\omega ) - F(t',\omega ')| < \varepsilon$.

ii.

We write $F \in \mathbb{B}(\Lambda _T)$ if for every $t_0 \in [0,T)$ and every $K>0$ there exists $C_{K,t_0}>0$ such that for all $t \in [0,t_0]$ and all $\omega \in D([0,T],\mathbb{R})$ with $\sup _{s \in [0,t]} |\omega (s)| \le K$ we have $|F(t,\omega )| \le C_{K,t_0}$.

iii.

$F$ is horizontally differentiable at $(t,\omega ) \in \Lambda _T$ if its horizontal derivative$$\begin{equation*} \mathcal{D} F(t,\omega ) :=\lim _{h\downarrow 0} \frac{F(t+h,\omega _t) - F(t,\omega _t)}{h} \end{equation*}$$

exists. If it exists for all $(t,\omega ) \in \Lambda _T$, then $\mathcal{D} F$ is a non-anticipative functional.

iv.

$F$ is vertically differentiable at $(t,\omega ) \in \Lambda _T$ if its vertical derivative$$\begin{equation*} \nabla _\omega F(t,\omega ) :=\lim _{h \downarrow 0} \frac{F(t,\omega _t + h \mathbf{1}_{[t,T]}) - F(t,\omega _t)}{h} \end{equation*}$$

exists. If it exists for all $(t,\omega ) \in \Lambda _T$, then $\nabla _\omega F$ is a non-anticipative functional. In particular, we define recursively $\nabla _\omega ^{k+1} F :=\nabla _\omega \nabla _\omega ^k F$ whenever this is well defined.

v.

For $p \in \mathbb{N}_0$ we say that $F \in \mathbb{C}^{1,p}_b(\Lambda _T)$ if $F$ is horizontally differentiable and $p$ times vertically differentiable in every $(t,\omega ) \in \Lambda _T$, and if $F, \mathcal{D} F, \nabla _\omega ^k F \in \mathbb{C}^{0,0}_l(\Lambda _T) \cap \mathbb{B}(\Lambda _T)$ for $k=1,\dots , p$.

Define the piecewise-constant approximation $S^n$ to $S$ along the partition $\pi _n$:

$$\begin{equation} S^n (t) = \sum _{[t_j, t_{j + 1}] \in \pi _n} S (t_{j + 1}) \mathbf{1}_{[t_j, t_{j + 1})} (t) + S (T) \mathbf{1}_{\{ T \}} (t) . {\tag{3}} \end{equation}$$

Then $\lim _{n \rightarrow \infty } \| S^n - S \|_{\infty } = 0$ whenever $\mathrm{osc}(S,\pi _n) \to 0$.

Theorem 1.10 (Functional change of variable formula for paths with finite $p$th variation).

Let $p$ be an even integer, let $F \in \mathbb{C}^{1, p}_b(\Lambda _T)$, and let $S \in V_p(\pi )$ for a sequence of partitions $(\pi _n)$ with vanishing mesh size $|\pi _n|\to 0$. Then the functional change of variable formula

$$\begin{align*} F (t, S_t) &= F (0, S_0) + \int _0^t \mathcal{D}F (s, S_s) \mathrm{d}s + \int _0^t \\ &<\nabla F (s, S_s), \mathrm{d}S (s)> + \frac{1}{p!} \int _0^t \nabla _{\omega }^p F (s, S_s) \mathrm{d}[S]^p (s) \end{align*}$$

holds, where

$$\begin{align*} \int _0^t &< \begin{aligned}[t] & \nabla F (s, S_s) ,\mathrm{d}S (s)> \\ & :=\lim _{n \rightarrow \infty } \sum _{[t_j, t_{j + 1}] \in \pi _n} \sum _{k=1}^{p-1} \frac{1}{k!} \nabla _{\omega }^k F (t_j, S^n_{t_j -}) (S (t_{j + 1} \wedge t) - S (t_j \wedge t))^k, \end{aligned} \end{align*}$$

with the piecewise constant approximation $S^n$ as defined in 3.

Proof.

Since the right hand side is a telescoping sum, we have

$$\begin{align*} F (t, S^n_t) - F (0, S^n_0) & = \sum _{[t_j, t_{j + 1}] \in \pi _n} (F (t_{j + 1} \wedge t, S^n_{(t_{j + 1} \wedge t) -}) - F (t_j \wedge t, S^n_{(t_j \wedge t) -})) \\ &\quad + F(t, S^n_t) - F(t, S^n_{t-}) \\ & = \sum _{[t_j, t_{j + 1}] \in \pi _n} (F (t_{j + 1} \wedge t, S^n_{(t_{j + 1} \wedge t) -}) \!-\! F (t_j \wedge t, S^n_{(t_j \wedge t) -})) \!+\! o (1). \end{align*}$$

Consider $j$ with $t_{j + 1} \leqslant t$ and split up the difference as follows:

$$\begin{equation*} F (t_{j + 1}, S^n_{t_{j + 1} -}) -\! F (t_j, S^n_{t_j -}) \!=\! (F (t_{j + 1}, S^n_{t_{j + 1} -}) - F (t_j, S^n_{t_j})) + (F (t_j, S^n_{t_j}) \!-\! F (t_j, S^n_{t_j -})) . \end{equation*}$$

Now $S^n_{t_{j + 1} -} (s) = S^n_{t_j} (s)$ for all $s \in [0, t_{j + 1}]$, and therefore the first term on the right hand side is simply

$$\begin{equation*} F (t_{j + 1}, S^n_{t_{j + 1} -}) - F (t_j, S^n_{t_j}) = \int _{t_j}^{t_{j + 1}} \mathcal{D}F (r, S^n_{t_j}) \mathrm{d}r, \end{equation*}$$

from where we easily get (using that the mesh size of $(\pi _n)$ converges to zero)

$$\begin{equation*} \lim _{n \rightarrow \infty } \sum _{[t_j, t_{j + 1}] \in \pi _n} (F (t_{j + 1} \wedge t, S^n_{(t_{j + 1} \wedge t) -}) - F (t_j \wedge t, S^n_{(t_j \wedge t)})) = \int _0^t \mathcal{D}F (r, S_r) \mathrm{d}r. \end{equation*}$$

It remains to consider the term

$$\begin{equation*} F (t_j, S^n_{t_j}) - F (t_j, S^n_{t_j -}) = F (t_j, S^{n, S_{t_j, t_{j + 1}}}_{t_j -}) - F (t_j, S^n_{t_j -}), \end{equation*}$$

where $S_{t_j, t_{j + 1}} :=S (t_{j + 1}) - S (t_j)$ and $S^{n, x}_{t_j-} (s) :=S^n_{t_j} (s) +\mathbf{1}_{[t_j,T]} (s) x$. By Taylor’s formula and the definition of the vertical derivative, we have

$$\begin{align*} \lefteqn{F (t_j, S^{n, S_{t_j, t_{j + 1}}}_{t_j -}) - F (t_j, S^n_{t_j -})}\qquad \\ &= \sum _{k=1}^p \frac{\nabla _{\omega }^k F (t_j, S_{t_j -}^n)}{k!} (S (t_{j + 1} \wedge t) - S (t_j \wedge t))^k\\ &\qquad {} + \frac{1}{(p - 1) !} \int _0^1 \mathrm{d}\lambda (1 - \lambda )^{p - 1} (S (t_{j + 1} \wedge t) - S(t_j \wedge t))^p \\ &\hspace{60pt} \times \left(\nabla _{\omega }^p F (t_j, S_{t_j -}^{n, \lambda S_{t_j, t_{j + 1}}}) - \nabla _{\omega }^p F (t_j, S_{t_j -}^n) \right). \end{align*}$$

Now we sum over $[t_j, t_{j + 1}] \in \pi _n$ and see as in Theorem 1.5 that the correction term vanishes for $n \rightarrow \infty$. Moreover, since $S \in V_p(\pi )$ we have

$$\begin{equation*} \lim _{n \to \infty } \sum _{[t_j, t_{j + 1}] \in \pi _n} \frac{\nabla _{\omega }^p F (t_j, S_{t_j -}^n)}{p!} (S (t_{j + 1} \wedge t) - S (t_j \wedge t))^p = \frac{1}{p!} \int _0^t \nabla _\omega ^p F(s,S_s) \mathrm{d}[S]^p(s); \end{equation*}$$

see 7, Lemma 5.3.7. Since $F \in \mathbb{C}^{0,0}_l(\Lambda _T)$, we have

$$\begin{equation*} \lim _{n \rightarrow \infty } (F (t, S^n_t) - F (0, S^n_0)) = F (t, S_t) - F (0, S_0), \end{equation*}$$

which completes the proof.

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2. Isometry relation and rough-smooth decomposition

Given a path (or process) $S \in V_p(\pi )$ with finite $p$th variation along the sequence of partitions $(\pi _n)$, the results above may be used to derive a decomposition of regular functionals of $S$ into a rough component with non-zero $p$th variation along $(\pi _n)$ and a smooth component with zero $p$th variation along $(\pi _n)$. For $p=2$ such a decomposition was obtained in 1 and is a pathwise analog of the decomposition of a Dirichlet process into a local martingale and a “zero energy” part 15.

For $\alpha \in (0,1)$ we write $C^\alpha ([0,T],\mathbb{R})$ for the $\alpha$-Hölder continuous paths from $[0,T]$ to $\mathbb{R}$, and $\|\cdot \|_\alpha$ denotes the $\alpha$-Hölder semi-norm.

2.1. An “isometry” property of the pathwise integral

Theorem 2.1 (“Isometry” formula).

Let $p \in \mathbb{N}$ be an even integer, let $\alpha > ((1+\tfrac{4}{p})^{1/2}-1)/2$, let $(\pi _n)$ be a sequence of partitions with mesh size going to zero, and let $S \in V_p(\pi ) \cap C^\alpha ([0,T],\mathbb{R})$. Let $F\in \mathbb{C}^{1,2}_b(\Lambda _T)$ such that $\nabla _\omega F \in \mathbb{C}^{1,1}_b(\Lambda _T)$. Assume furthermore that $F$ is Lipschitz-continuous with respect to $d_\infty$. Then $F(\cdot , S) \in V_p(\pi )$ and

$$\begin{equation*} [F(\cdot ,S)]^p(t)=\int _0^t |\nabla _\omega F(s,S_s)|^p \mathrm{d}[S]^p(s). \end{equation*}$$

Proof.

The proof is similar to the case $p=2$ considered in 1. Indeed, our assumptions allow us to apply 1, Lemma 2.2, which shows that there exists $C>0$, only depending on $T$, $F$, and $\|S\|_\alpha$, such that for all $0 \le s \le t \le T$

$$\begin{equation} |R_F(s,t)| :=|F(t,S_t) - F(s, S_s) - \nabla _\omega F(s,S_s) (S(t) - S(s))| \le C |t-s|^{\alpha +\alpha ^2}. {\tag{4}} \end{equation}$$

Writing also $\gamma _F(s,t) :=\nabla _\omega F(s,S_s) (S(t) - S(s))$, we obtain

$$\begin{equation} \begin{gathered} \sum _{\substack{[t_j, t_{j + 1}] \in \pi _n:\\ t_{j+1} \le t}} |F(t_{j+1}, S_{t_{j + 1}}) - F(t_j, S_{t_j})|^p = \sum _{\substack{[t_j, t_{j + 1}] \in \pi _n:\\ t_{j+1} \le t}} |R_F(t_j, t_{j+1}) + \gamma _F(t_j,t_{j+1})|^p \\ = \sum _{\substack{[t_j, t_{j + 1}] \in \pi _n:\\ t_{j+1} \le t}} |\gamma _F(t_j,t_{j+1})|^p + \sum _{k=1}^p \binom{p}{k} \sum _{\substack{[t_j, t_{j + 1}] \in \pi _n:\\ t_{j+1} \le t}} R_F(t_j, t_{j+1})^k \gamma _F(t_j,t_{j+1})^{p-k}. \end{gathered} {\tag{5}} \end{equation}$$

Since $S \in V_p(\pi )$ we have

$$\begin{equation} \lim _{n \to \infty } \sum _{\substack{[t_j, t_{j + 1}] \in \pi _n:\\ t_{j+1} \le t}} |\gamma _F(t_j, t_{j+1})|^{p} = \int _0^t |\nabla _\omega F(s,S(s))|^p \mathrm{d}[S]^p(s). {\tag{6}} \end{equation}$$

Our result follows once we show that the double sum on the right hand side of 5 vanishes. For that purpose let $k \in \{1,\dots ,p\}$ and write $q_k :=p/(p-k) \in [1,\infty ]$ and let $q_k' = p/k$ be its conjugate exponent. Hölder’s inequality yields

$$\begin{align*} & \Big | \sum _{\substack{[t_j, t_{j + 1}] \in \pi _n:\\ t_{j+1} \le t}} R_F(t_j, t_{j+1})^k \gamma _F(t_j,t_{j+1})^{p-k} \Big | \\ & \le \Big ( \sum _{\substack{[t_j, t_{j + 1}] \in \pi _n:\\ t_{j+1} \le t}} |R_F(t_j, t_{j+1})|^{k q_k'} \Big )^{1/q_k'} \Big ( \sum _{\substack{[t_j, t_{j + 1}] \in \pi _n:\\ t_{j+1} \le t}} |\gamma _F(t_j, t_{j+1})|^{(p-k) q_k} \Big )^{1/q_k} \\ & = \Big ( \sum _{\substack{[t_j, t_{j + 1}] \in \pi _n:\\ t_{j+1} \le t}} |R_F(t_j, t_{j+1})|^{p} \Big )^{k/p} \Big ( \sum _{\substack{[t_j, t_{j + 1}] \in \pi _n:\\ t_{j+1} \le t}} |\gamma _F(t_j, t_{j+1})|^{p} \Big )^{(p-k)/p}. \end{align*}$$

By 4 the first sum on the right hand side is bounded by

$$\begin{align*} &\Big ( \sum _{\substack{[t_j, t_{j + 1}] \in \pi _n:\\ t_{j+1} \le t}} |R_F(t_j, t_{j+1})|^{p} \Big )^{k/p}\\ &\qquad \begin{aligned}[t] & \lesssim \Big ( \sum _{\substack{[t_j, t_{j + 1}] \in \pi _n:\\ t_{j+1} \le t}} |t_{j+1} - t_j|^{p(\alpha +\alpha ^2)} \Big )^{k/p}\\ & \le (t \times \max \{|t_{j+1} - t_j|^{p(\alpha + \alpha ^2) - 1}: [t_j,t_{j+1}] \in \pi _n, t_{j+1} \le t\})^{k/p}, \end{aligned} \end{align*}$$

which converges to zero for $n \to \infty$ because $p(\alpha +\alpha ^2) > 1$ (which is equivalent to our assumption $\alpha > (\sqrt {1+\frac{4}{p}}-1)/2$) and because $k > 0$. Moreover, by 6 the sum over $|\gamma _F(t_j, t_{j+1})|^p$ is bounded and this concludes the proof.

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Remark 2.2.

(1)

Keeping the example of the (fractional) Brownian motion in mind, we would typically expect paths in $V_p(\pi )$ to be $(1/p-\kappa )$-Hölder continuous for any $\kappa > 0$. Since for $f(x) = (1+x)^{1/2}$ we have$$\begin{equation*} f''(x) = -\frac{1}{4} (1+x)^{-3/2} < 0, \end{equation*}$$

we have $f(x) < f(0) + f'(0) x$ for all $x > 0$, and therefore$$\begin{equation*} \frac{(1+\frac{4}{p})^{1/2}-1}{2} < \frac{\frac{1}{2}\frac{4}{p}}{2} = \frac{1}{p}, \end{equation*}$$

which means that in Theorem 2.1 we can take $\alpha < 1/p$ and our constraint on the Hölder regularity is not unreasonable.

(2)

In fact the constraint on $\alpha$ comes from inequality 4, which only gives us a control of order $|t-s|^{\alpha +\alpha ^2}$ for $R_F(s,t)$, while $|t-s|^{2\alpha }$ might seem more natural (after all $R_F(s,t)$ is something like the remainder in a first-order Taylor expansion). The difficulty is that horizontal differentiability is a very weak notion and gives us no control on $R_F(s,t)$. To obtain any bounds at all we first need to approximate our path by piecewise linear or piecewise constant paths, and through this approximation procedure we lose a little bit of regularity; see 1, Lemma 2.2 for details. One can improve the estimate on $R_F(s,t)$ by taking a higher-order Taylor expansion (which would require more regularity from $F$), but we do not need this here.

2.2. Pathwise rough-smooth decomposition

Using the above result we may derive, as in 1, a pathwise “signal plus noise” decomposition for regular functionals of paths with strictly increasing $p$th variation. Let

$$\begin{equation*} \mathbb{C}^{1,p}_b(S)=\{ F(\cdot ,S) , F\in \mathbb{C}^{1,p}_b(\Lambda _T) \}\subset V_p(\pi ). \end{equation*}$$

The following result extends the pathwise rough-smooth decomposition of paths in $\mathbb{C}^{1,p}_b(S)$, obtained in 1 for $p=2$, to higher values of $p$.

Theorem 2.3 (Rough-smooth decomposition).

Let $p \in \mathbb{N}$ be an even integer, let $\alpha >((1+\tfrac{4}{p})^{1/2}-1)/2$, let $(\pi _n)$ be a sequence of partitions with vanishing mesh size $|\pi _n|\to 0$ and let $S\in V_p(\pi ) \cap C^\alpha ([0,T],\mathbb{R})$ be a path with strictly increasing $p$th variation $[S]^p$ along $(\pi _n)$. Then any $X\in \mathbb{C}^{1,p}_b(S)$ admits a unique decomposition

$$\begin{equation*} X= X(0)+ A+M\quad \text{where}\quad [A]^p = 0\quad \text{and}\quad M(t)= \int _0^t \phi (s)\mathrm{d}S(s) \end{equation*}$$

is a pathwise integral defined as in Theorem 1.10.

Proof.

Existence of the decomposition is a consequence of Theorem 1.10. Consider two such decompositions $X - X_0 = A + M = \tilde{A} + \tilde{M}$. Since $[A]^p = [\tilde{A}]^p = 0$ and

$$\begin{equation*} |(A - \tilde{A})(t) - (A - \tilde{A})(s)|^p \lesssim |A(t) - A(s)|^p + |\tilde{A}(t) - \tilde{A}(s)|^p, \end{equation*}$$

we get $A - \tilde{A} \in V_p(\pi )$ and $[A- \tilde{A}]^p \equiv 0$. But then also $[M - \tilde{M}]^p = [A - \tilde{A}]^p \equiv 0$. Now

$$\begin{equation*} M(t) = \int _0^t \nabla _\omega F(s, S_s) \mathrm{d}S(s),\qquad \tilde{M}(t) = \int _0^t \nabla _\omega \tilde{F}(s,S_s) \mathrm{d}S(s) \end{equation*}$$

for some $F, \tilde{F} \in C^{1,p}_b(\Lambda _T)$, and by Theorem 2.1 we have

$$\begin{equation*} 0 = [M - \tilde{M}]^p(T) = \int _0^T | \nabla _\omega (F - \tilde{F})(s,S_s)|^p \mathrm{d}[S]^p(s). \end{equation*}$$

Since $(F - \tilde{F})(s,S_s)$ is continuous in $s$ and $[S]^p$ is strictly increasing we have $\nabla _\omega (F - \tilde{F})(\cdot , S) \equiv 0$. This means that $M - \tilde{M} \equiv 0$, and then also $A - \tilde{A} \equiv 0$.

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Remark 2.4.

If $t\mapsto [S]^p(t)$ is not strictly increasing, uniqueness of the decomposition still holds $d[S]^p-$almost everywhere.

3. Local times and higher-order Wuermli formula

An extension of Föllmer’s pathwise Itô formula to less regular functions was given by Wuermli 30 in her (unpublished) thesis. Wuermli considered paths with finite quadratic variation which further admit a local time along a sequence of partitions, and derive a pathwise change of variable formula for more general functions that need not be $C^2$. Depending on the notion of convergence used to define the local time, one then obtains a Tanaka-type change of variable formulas for various classes of functions; convergence in stronger topologies leads to a formula valid for a larger class of functions. Wuermli 30 assumed weak convergence in $L^2$ in the space variable (see also 2) and some recent works have extended the approach to other topologies, for example uniform convergence or weak convergence in $L^q$ 1024. To a certain extent Wuermli’s approach can be generalized to our higher-order setting, but as we will discuss below in the higher-order case we do not expect to have convergence of the pathwise local times in strong topologies.

To derive the generalization of Wuermli’s formula, we consider $f \in C^{p-2}$ with absolutely continuous $f^{(p-2)}$ and apply the Taylor expansion of order $p-2$ with integral remainder to obtain

$$\begin{equation*} f (b) - f (a) = \sum _{k=1}^{p-2} \frac{f^{(k)}(a)}{k!}(b-a)^k + \int _a^b \frac{f^{(p-1)}(x)}{(p-2)!} (b-x)^{p-2} \mathrm{d}x. \end{equation*}$$

Assume now that $f^{(p-1)}$ is of bounded variation. Since every bounded variation function $f^{(p-1)}$ is regulated (làdlàg) and therefore has only countably many jumps, its càdlàg version is also a weak derivative of $f^{(p-2)}$, and from now on we only work with this version. Since $(b-\cdot )^{p-2}$ is continuous, the integration by parts rule for the Lebesgue-Stieltjes integral applies in the case $b\ge a$ and we obtain

$$\begin{align*} \int _a^b \frac{f^{(p-1)}(x)}{(p-2)!} (b-x)^{p-2} \mathrm{d}x & = f^{(p-1)}(b) \frac{-(b-b)^{p-1}}{(p-1)!} - f^{(p-1)}(a)\frac{-(b-a)^{p-1}}{(p-1)!} \\ &\quad - \int _{(a,b]} \frac{-(b-x)^{p-1}}{(p-1)!} \mathrm{d}f^{(p-1)}(x) \\ & = f^{(p-1)}(a)\frac{(b-a)^{p-1}}{(p-1)!} + \int _{(a,b]} \frac{(b-x)^{p-1}}{(p-1)!} \mathrm{d}f^{(p-1)}(x). \end{align*}$$

Similarly we get for $b < a$

$$\begin{align*} \int _a^b \frac{f^{(p-1)}(x)}{(p-2)!} (b-x)^{p-2} \mathrm{d}x & = - \int _b^a \frac{f^{(p-1)}(x)}{(p-2)!} (b-x)^{p-2} \mathrm{d}x \\ & = f^{(p-1)}(a)\frac{(b-a)^{p-1}}{(p-1)!} - \int _{(b,a]} \frac{(b-x)^{p-1}}{(p-1)!} \mathrm{d}f^{(p-1)}(x), \end{align*}$$

and therefore

$$\begin{align*} f (b) - f (a) & = \sum _{k=1}^{p-1} \frac{f^{(k)}(a)}{k!}(b-a)^k + \operatorname {sign}(b-a)\int _{\llparenthesis a,b\rBrack } \frac{(b-x)^{p-1}}{(p-1)!} \mathrm{d}f^{(p-1)}(x) \\ & = \sum _{k=1}^{p-1} \frac{f^{(k)}(a)}{k!}(b-a)^k + \operatorname {sign}(b-a)^{p} \int _{\llparenthesis a,b\rBrack } \frac{|b-x|^{p-1}}{(p-1)!} \mathrm{d}f^{(p-1)}(x) \\ & = \sum _{k=1}^{p-1} \frac{f^{(k)}(a)}{k!}(b-a)^k + \int _{\mathbb{R}} \mathbf{1}_{\llparenthesis a,b \rBrack }(x) \frac{\operatorname {sign}(b-a)^p|b-x|^{p-1}}{(p-1)!} \mathrm{d}f^{(p-1)}(x), \end{align*}$$

with the notation

$$\begin{equation*} \llparenthesis a,b \rBrack = \begin{cases} (a,b], & b \ge a, \\ (b,a], & a \le b. \end{cases} \end{equation*}$$

For any partition $\sigma$ of $[0, T]$, we define

$$\begin{equation*} L^{\sigma , p - 1}_t(x) :=\sum _{t_j \in \sigma } \operatorname {sign}(S_{t_{j+1} \wedge t} - S(t_j \wedge t))^p \mathbf{1}_{\llparenthesis S(t_j \wedge t), S_{t_{j + 1} \wedge t}\rBrack } (x) | S(t_{j + 1} \wedge t) - x |^{p - 1}. \end{equation*}$$

To extend Theorem 1.5 to $S\in V_p(\pi )$, we first note that the following identity holds for any partition $\pi _n$:

$$\begin{align} f (S_t) - f (S_0) & = \sum _{[t_j, t_{j + 1}] \in \pi _n} \sum _{k=1}^{p-1} \frac{f^{(k)} (S_{t_j})}{k!} (S(t_{j + 1} \wedge t) - S(t_j \wedge t))^k \\ &\quad + \frac{1}{(p-1)!} \int _{\mathbb{R}} L^{\pi _n, p - 1}_t (x) \mathrm{d}f^{(p - 1)} (x) . {\tag{7}} \end{align}$$

To obtain a change of variable formula for less regular functions, we need the last term to converge as the partition is refined. This motivates the following definition.

Definition 3.1 (Local time of order $p$).

Let $p\in \mathbb{N}$ be an even integer and let $q \in [1,\infty ]$. A continuous path $S \in C([0,T],\mathbb{R})$ has an $L^q$-local time of order $p-1$ along a sequence of partitions $\pi =(\pi _n)_{n\geq 1}$ if $\operatorname {osc}(S,\pi _n)\to 0$ and

$$\begin{equation*} L^{\pi _n, p - 1}_t(\cdot ) = \sum _{t_j \in \pi } \mathbf{1}_{\llparenthesis S(t_j \wedge t), S_{t_{j + 1} \wedge t}\rBrack } (\cdot ) | S(t_{j + 1} \wedge t) - \cdot |^{p - 1} \end{equation*}$$

converges weakly in $L^q(\mathbb{R})$ to a weakly continuous map $L\colon [0,T]\to L^q(\mathbb{R})$ which we call the order $p$ local time of $S$. We denote $\mathcal{L}_p^q(\pi )$ the set of continuous paths $S$ with this property.

Intuitively, the limit $L_t(x)$ then measures the rate at which the path $S$ accumulates $p$th order variation near $x$. This definition is further justified by the following result, which is a ‘pathwise Tanaka formula’ 30 for paths of arbitrary regularity.

Theorem 3.2 (Pathwise “Tanaka” formula for paths with finite $p$th order variation).

Let $p \in 2\mathbb{N}$ be an even integer, $q \in [1,\infty ]$ with conjugate exponent $q'=q/(q-1)$. Let $f \in C^{p-1}(\mathbb{R},\mathbb{R})$ and assume that $f^{(p-1)}$ is weakly differentiable with derivative in $L^{q'}(\mathbb{R})$. Then for any $S \in \mathcal{L}^q_p(\pi )$ the pointwise limit of compensated Riemann sums

$$\begin{equation*} \int _0^t f' (S(s)) \mathrm{d}S(s) :=\lim _{n \rightarrow \infty } \sum _{[t_j, t_{j + 1}] \in \pi _n} \sum _{k=1}^{p-1} \frac{f^{(k)} (S(t_j))}{k!} (S(t_{j + 1} \wedge t) - S(t_j \wedge t))^k \end{equation*}$$

exists and the following change of variable formula holds:

$$\begin{equation*} f(S(t)) - f(S(0)) = \int _0^t f' (S(s)) \mathrm{d}S(s) + \frac{1}{(p-1)!} \int _\mathbb{R}f^{(p)}(x) L_t(x) \mathrm{d}x. \end{equation*}$$

Proof.

The formula 7 is exact and does not involve any error terms. Noting that $L^{q'}(\mathbb{R}) \subset (L^q)^\ast (\mathbb{R})$ also for $q=\infty$, our assumptions imply that the second term on the right hand side of 7 converges, so the result follows.

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To justify the name “local time” for $L$, we illustrate how $L$ is related to classical definitions of local times by restricting our attention to a particular sequence of partitions 620.

Definition 3.3.

Let $S \in C([0,T], \mathbb{R})$. The dyadic Lebesgue partition generated by $S$ is defined via $\tau ^n_0 :=0$ and

$$\begin{equation*} \tau ^n_{j+1} :=\inf \{ t \ge \tau ^n_j: S_t \in 2^{-n} \mathbb{Z}\setminus \{ S_{\tau ^n_j}\} \}, \end{equation*}$$

and then $\pi _n = (\{\tau ^n_j : j \in \mathbb{N}_0\} \cap [0,T]) \cup \{T\}$.

Lemma 3.4.

Let $p \in \mathbb{N}$ be even, let $S \in C([0,T], \mathbb{R})$, and let $(\pi _n)$ be the dyadic Lebesgue partition generated by $S$. Given an interval $[a,b]$ we write $U_t([a,b])$ for the number of upcrossings of $[a,b]$ that $S$ performs until time $t$. Let $x \in \mathbb{R}$ and let $I^n_k = (k 2^{-n},(k+1)2^{-n}]$ be the unique dyadic interval of generation $n$ with $x \in I^n_k$. Then

$$\begin{equation*} L^{\pi _n}_t(x) = (|(k+1)2^{-n} - x|^{p-1} + |x - k2^{-n}|^{p-1})U_t(I^n_k) + O(2^{-n(p-1)}). \end{equation*}$$

Proof.

We have $\mathbf{1}_{\llparenthesis S_{\tau _j^n}, S_{\tau ^n_{j + 1}}\rBrack } (x) \neq 0$ if either $S_{\tau ^n_j} = k 2^{-n}$ and $S_{\tau ^n_{j+1}} = (k+1) 2^{-n}$ (i.e., $S$ performs an upcrossing of $I^n_k$), or $S_{\tau ^n_j} = (k+1) 2^{-n}$ and $S_{\tau ^n_{j+1}} = k 2^{-n}$ (i.e., $S$ performs a downcrossing of $I^n_k$). In the first case we have to add $|(k+1)2^{-n} - x|^{p-1}$ to $L^{\pi _n}_t(x)$, and in the second case we add $(-1)^p | x - k 2^{-n}|^{p-1} = | x - k 2^{-n}|^{p-1}$. Therefore, we obtain

$$\begin{equation*} L^{\pi _n}_t(x) = |(k+1)2^{-n} - x|^{p-1} U_t(I^n_k) + |x - k2^{-n}|^{p-1} D_t(I^n_k) + O(2^{-n(p-1)}), \end{equation*}$$

and since up- and downcrossings of $I^n_k$ differ by at most one, our claim follows.

■

Note that the expression for $L^{\pi _n}_t$ strongly fluctuates on $I^n_k$. For $x \simeq k2^{-n}$ and $x \simeq (k+1) 2^{-n}$ the factor in front of $U_t(I^n_k)$ is $\simeq 2^{-n(p-1)}$, while for $x = (2k+1) 2^{-n-1}$ we get the factor $2^{-n(p-1)} 2^{p-2}$. Therefore, we do not expect $L^{\pi _n}_t(x)$ to converge uniformly or even pointwise in $x$ as $n\to \infty$ (unless if $p=2$).

Lemma 3.5.

In the setting of Lemma 3.4 set

$$\begin{equation*} \tilde{L}^{\pi _n}_t(x) :=\sum _{k \in \mathbb{Z}} 2^{-n(p-1)} U_t(I^n_k) \mathbf{1}_{I^n_k}(x). \end{equation*}$$

Let $q \in (1,\infty )$. If $\tilde{L}^{\pi _n}_t$ converges weakly in $L^q(\mathbb{R})$ to a limit $\tilde{L}_t$, then $L^{\pi _n}_t$ converges weakly in $L^q(\mathbb{R})$ to $(2/p) \tilde{L}_t$.

Proof.

Let us introduce an averaging operator,

$$\begin{equation*} (\mathcal{A}_n f)(x) :=\sum _{k \in \mathbb{Z}} 2^n \int _{I^n_k} f(y) \mathrm{d}y \, \mathbf{1}_{I^n_k}(x). \end{equation*}$$

Since

$$\begin{equation*} \int _{I^n_k} (|(k+1)2^{-n} - x|^{p-1} + |x - k2^{-n}|^{p-1}) \mathrm{d}x = 2 \int _0^{2^{-n}} x^{p-1} \mathrm{d}x = \frac{2}{p} 2^{-n p}, \end{equation*}$$

we have $\tilde{L}^{\pi _n}_t = \tfrac{p}{2} \mathcal{A}_n L^{\pi _n}_t + O(2^{-n(p-1)})$, with a compactly supported remainder $O(2^{-n(p-1)})$. We claim that if $(f_n)$ is a sequence of functions for which $\mathcal{A}_n f_n$ converges weakly in $L^q(\mathbb{R})$ and for which $|f_n| \le C |\mathcal{A}_n f_n|$, then also $(f_n)$ converges weakly in $L^q(\mathbb{R})$ to the same limit, which will imply our claim. To show this, let $f$ be the limit of $\mathcal{A}_n$ and let $g \in L^{q'}(\mathbb{R})$. We have $\langle \mathcal{A}_n \varphi , \psi \rangle = \langle \mathcal{A}_n \varphi , \mathcal{A}_n \psi \rangle = \langle \varphi , \mathcal{A}_n \psi \rangle$ for all $\varphi , \psi$, and therefore

$$\begin{align*} |\langle f_n - f, g \rangle | & \le |\langle f_n - \mathcal{A}_n f_n, g \rangle | + |\langle \mathcal{A}_n f_n - f, g \rangle | \\ & = |\langle f_n, g - \mathcal{A}_n g \rangle | + |\langle \mathcal{A}_n f_n - f, g \rangle | \\ & \le \| f_n \|_{L^q} \| g - \mathcal{A}_n g \|_{L^{q'}} + |\langle \mathcal{A}_n f_n - f, g \rangle |. \end{align*}$$

The second term on the right hand side converges to zero by assumption. For the first term we note that by assumption $\| f_n \|_{L^q} \le \| \mathcal{A}_n f_n \|_{L^q}$, which is uniformly bounded in $n$ because $(\mathcal{A}_n f_n)$ converges weakly in $L^q$. The proof is therefore complete once we show that $\lim _{n \to \infty } \| g - \mathcal{A}_n g \|_{L^{q'}} = 0$ for all $g \in L^{q'}$. But this easily follows from the fact that the continuous and compactly supported functions are dense in $L^{q'}$.

■

In fact, we conjecture that, for fractional Brownian motion, this notion of local time defined along the dyadic Lebesge partition coincides, up to a constant, with the usual concept of local time defined as the density of the occupation measure.

Conjecture.

Let $B$ be the fractional Brownian motion with Hurst parameter $H \in (0,1)$, and let $(\pi _n)$ be the dyadic Lebesgue partition generated by $B$. Let $I^n_k$ and $U_t$ be as in Lemma 3.4 (where now we count the upcrossings of $B$ instead of $S$). We conjecture that

$$\begin{equation*} \tilde{L}^{\pi _n}_t(x) :=\sum _{k \in \mathbb{Z}} 2^{-n(1/H-1)} U_t(I^n_k) \mathbf{1}_{I^n_k}(x) \end{equation*}$$

almost-surely converges uniformly in $(t,x) \in [0,T] \times \mathbb{R}$ to $\ell _t(x) \mathbb{E}[|B_1|^{1/H}]/2$, where $\ell$ is the local time of $B$, i.e., the Radon-Nikodym derivative of the occupation measure $A \mapsto \int _0^t \mathbf{1}_A(B(s)) \mathrm{d}s$ with respect to the Lebesgue measure; see, e.g., 3. In particular, for any even integer $p\in 2\mathbb{N}$, $B \in \mathcal{L}^{p-1}_q(\pi _n)$ for any $q \in (1,\infty )$.

This result is well known for $H = 1/2$; see, e.g., 624. In the general case $H \in (0,1)$, it is natural to expect that

$$\begin{equation*} \mu ^n([0,t]) :=\sum _{j=0}^\infty 2^{-n/H} \mathbf{1}_{\tau ^n_{j+1} \le t} \xrightarrow {n \to \infty } [B]^{1/H}_t = \mathbb{E}[|B_1|^{1/H}] t, \end{equation*}$$

which would be an extension of the convergence result of 27 from deterministic partitions to the Lebesgue partition generated by $B$. Moreover, we know that the local time $\ell$ of the fractional Brownian motion satisfies

$$\begin{equation*} \ell _t(x) = \lim _{n \to \infty } \sum _{k \in \mathbb{Z}} 2^n \int _0^t \mathbf{1}_{I^n_k}(B_s) \mathrm{d}s \mathbf{1}_{I^n_k}(x). \end{equation*}$$

If we formally replace the Lebesgue measure in the integral by $\mathbb{E}[|B_1|^{1/H}]^{-1} \mu ^n$, then we get

$$\begin{align*} \ell _t(x) & = \mathbb{E}[|B_1|^{1/H}]^{-1}\lim _{n \to \infty } \sum _{k \in \mathbb{Z}} 2^n \int _0^t \mathbf{1}_{I^n_k}(B_s) \mu ^n(\mathrm{d}s) \mathbf{1}_{I^n_k}(x) \\ & = \mathbb{E}[|B_1|^{1/H}]^{-1}\lim _{n \to \infty } \sum _{k \in \mathbb{Z}} 2^{n - n/H} \sum _{j: \tau ^n_{j+1} \le t} \mathbf{1}_{I^n_k}(B_{\tau ^n_j}) \mathbf{1}_{I^n_k}(x) \\ & = \mathbb{E}[|B_1|^{1/H}]^{-1}\lim _{n \to \infty } \sum _{k \in \mathbb{Z}} 2^{n - n/H} (D_t(I^n_k) + U_t(I^n_{k+1})) \mathbf{1}_{I^n_k}(x), \end{align*}$$

and if we further assume that $2^{n - n/H} |U_t(I^n_{k+1}) - U_t(I^n_k)|\to 0$, then our conjecture formally follows.

If the conjecture holds, then for any $p\in 2\mathbb{N}$ and $B$ a typical sample path of the fractional Brownian motion with Hurst index $1/p$ and $f \in C^{p-1}$ with weak $p$th derivative $f^{(p)} \in L^q$ for any $q \in (1,\infty )$:

$$\begin{equation} f(B(t)) - f(B(0)) = \int _0^t f' (B(s)) \mathrm{d}B(s) + \frac{\mathbb{E}[|B_1|^p]}{p!} \int _\mathbb{R}f^{(p)}(x) \ell _t(x) \mathrm{d}x, {\tag{8}} \end{equation}$$

where $\ell$ is the local time of $B$ and

$$\begin{equation*} \int _0^t f' (B(s)) \mathrm{d}B(s) :=\lim _{n \rightarrow \infty } \sum _{[t_j, t_{j + 1}] \in \pi _n} \sum _{k=1}^{p-1} \frac{f^{(k)} (B(t_j))}{k!} (B(t_{j + 1} \wedge t) - B(t_j \wedge t))^k. \end{equation*}$$

By Theorem 1.5 the formula holds for $f \in C^p$, because then

$$\begin{equation*} \frac{\mathbb{E}[|B_1|^p]}{p!} \int _\mathbb{R}f^{(p)}(x) \ell _t(x) \mathrm{d}x = \frac{\mathbb{E}[|B_1|^p]}{p!} \int _0^t f^{(p)}(S(s)) \mathrm{d}s = \frac{1}{p!} \int _0^t f^{(p)}(S(s)) \mathrm{d}[S]^p_s, \end{equation*}$$

which adds further credibility to our conjecture.

4. Extension to multidimensional paths

As in the case $p=2$, the set $V_p(\pi )$ is not stable under linear combinations: for $S_1,S_2\in V_p(\pi )$, expanding $( (S_1(t_{j + 1}) - S_1(t_j)+ S_2(t_{j + 1}) - S_2(t_j))^p$ yields many cross terms whose sum cannot be controlled in general as the partition is refined. The extension of Definition 1.1 to vector-valued functions $S=(S_1,\ldots ,S_d)$ therefore requires some care. The original approach of Föllmer 14 was to require that $S_i, S_i+S_j \in V_p(\pi )$. We propose here a slightly different formulation, which is equivalent to Föllmer’s construction for $p=2$ but easier to relate to other approaches, such as rough path integration.

4.1. Tensor formulation

Define $T_p(\mathbb{R}^d) =\mathbb{R}^d\otimes \ldots \otimes \mathbb{R}^d$ as the space of $p$-tensors on $\mathbb{R}^d$. A symmetric $p$-tensor is a tensor $T\in T_p(\mathbb{R}^d)$ that is invariant under any permutation $\sigma$ of its arguments:

$$\begin{equation*} \forall (v_{1},v_{2},\ldots ,v_{p})\in (\mathbb{R}^d)^p,\qquad {\displaystyle T(v_{1},v_{2},\ldots ,v_{p})=T(v_{\sigma 1},v_{\sigma 2},\ldots ,v_{\sigma p})}. \end{equation*}$$

The coordinates $(T_{i_{1}i_{2}\cdots i_{p}})$ of a symmetric tensor of order $p$ satisfy

$$\begin{equation*} {\displaystyle T_{i_{1}i_{2}\cdots i_{p}}=T_{i_{\sigma 1}i_{\sigma 2}\cdots i_{\sigma p}}.} \end{equation*}$$

The space $\operatorname {Sym}_p(\mathbb{R}^d)$ of symmetric tensors of order $p$ on $\mathbb{R}^d$ is naturally isomorphic to the dual of the space $\mathbb{H}_p[X_1,\ldots ,X_d]$ of symmetric homogeneous polynomials of degree $p$ on $\mathbb{R}^d$. We set $\operatorname {Sym}_0(\mathbb{R}^d):=\mathbb{R}.$

An important example of a symmetric $p$-tensor on $\mathbb{R}^d$ is given by the $p$th order derivative of a smooth function:

$$\begin{equation*} \forall f\in C^p(\mathbb{R}^d,\mathbb{R}), \forall x\in \mathbb{R}^d: \qquad \nabla ^p f(x)\in \operatorname {Sym}_p(\mathbb{R}^d). \end{equation*}$$

The symmetry property is obtained by repeated application of Schwarz’s lemma.

We define $\mathbb{S}_p(\mathbb{R}^d)$ as the direct sum of $\operatorname {Sym}_k(\mathbb{R}^d)$ for $k = 0,1,2,\ldots ,p$:

$$\begin{equation*} \mathbb{S}_p(\mathbb{R}^d) =\bigoplus _{k=0}^{p} \operatorname {Sym}_{k}(\mathbb{R}^d). \end{equation*}$$

The space $\mathbb{S}_p(\mathbb{R}^d)$ is naturally isomorphic to the dual of the space $\mathbb{R}_p[X_1,\ldots ,X_d]$ of polynomials of degree $\leq p$ in $d$ variables, which defines a bilinear product

$$\begin{equation*} \langle \cdot ,\cdot \rangle \colon \mathbb{S}_p(\mathbb{R}^d)\times \mathbb{R}_p[X_1,\ldots ,X_d] \to \mathbb{R}. \end{equation*}$$

Slightly abusing notation, we also write $\langle \cdot , \cdot \rangle$ for the canonical inner product on $T_p(\mathbb{R}^d)$. Consider now a continuous $\mathbb{R}^d$-valued path $S\in C([0,T],\mathbb{R}^d)$ and a sequence of partitions $\pi _n\!=\!\{t_0^n, \dots , t^n_{N(\pi _n)}\}$ with $t_0^n\!=\!0<\ldots < t^n_k<\ldots < t^n_{N(\pi _n)}\!=\!T$. Then

$$\begin{equation*} \mu ^n :=\sum _{[t_j,t_{j+1}] \in \pi _n} \delta (\cdot - t_j) \underbrace{(S(t_{j + 1}) - S(t_j))\otimes \dots \otimes (S(t_{j + 1}) - S(t_j))}_{\text{$p$ times}} \end{equation*}$$

defines a tensor-valued measure on $[0,T]$ with values in $\operatorname {Sym}_p(\mathbb{R}^d)$. This space of measures is in duality with the space $C([0,T],\mathbb{H}_p[X_1,\ldots ,X_d] )$ of continuous functions taking values in homogeneous polynomials of degree $p$, i.e., homogeneous polynomials of degree $p$ with continuous time-dependent coefficients.

Definition 4.1 ($p$th variation of a multidimensional function).

Let $p\in \mathbb{N}$ be even, let $S \in C([0,T],\mathbb{R}^d)$ be a continuous path, and let $\pi =(\pi _n)_{n\geq 1}$ be a sequence of partitions of $[0,T]$. Consider the sequence of tensor-valued measures

$$\begin{equation*} \mu ^n :=\sum _{[t_j,t_{j+1}] \in \pi _n} \delta (\cdot - t_j) (S(t_{j + 1}) - S(t_j))^{\otimes p}. \end{equation*}$$

We say that $S$ has a $p$th variation along $\pi =(\pi _n)_{n\geq 1}$ if $\operatorname {osc}(S,\pi _n)\to 0$ and there exists a $\operatorname {Sym}_p(\mathbb{R}^d)$–valued measure $\mu _S$ without atoms such that for all $f\in C([0,T], \mathbb{H}_p[X_1,\ldots ,X_d])$

$$\begin{align*} \lim _{n\to \infty } \int _{0}^T\langle f,\mathrm{d}\mu _n \rangle = \lim _{n\to \infty } \sum _{[t_j,t_{j+1}] \in \pi _n} \langle f( t_j), (S(t_{j + 1}) - S(t_j))^{\otimes p} \rangle = \int _{0}^T\langle f,\mathrm{d}\mu _S\rangle . \end{align*}$$

In that case we write $S \in V_p(\pi )$ and we call $[S]^p\colon [0,T]\to \operatorname {Sym}_p(\mathbb{R}^d)$ defined by

$$\begin{equation*} [S]^p(t) := \mu ([0,t]) \end{equation*}$$

the $p$th variation of $S$.

By analogy with the positivity property of symmetric matrices, we say that a symmetric $p$-tensor $T \in \operatorname {Sym}_p(\mathbb{R}^d)$ is positive if

$$\begin{align*} \langle T,v\otimes \ldots \otimes v \rangle \geq 0 \qquad \forall v\in \mathbb{R}^d. \end{align*}$$

We denote the set of positive symmetric $p$-tensors by $\operatorname {Sym}_p^+(\mathbb{R}^d)$. For $T,\tilde{T}\in \operatorname {Sym}_p(\mathbb{R}^d)$ we write $T\geq \tilde{T}$ if $T-\tilde{T}\in \operatorname {Sym}^+_p(\mathbb{R}^d)$. This defines a partial order on $\operatorname {Sym}_p(\mathbb{R}^d)$.

Property 4.2.

Let $S \in V_p(\pi )\cap C([0,T],\mathbb{R}^d)$. Then

(i)

$[S]^p$ has finite variation and is increasing in the sense of the partial order on $\operatorname {Sym}_p(\mathbb{R}^d)$:$$\begin{equation*} [S]^p(t+h) - [S]^p(t) \in \operatorname {Sym}^+_p(\mathbb{R}^d) \qquad \forall \, 0\leq t\leq t+h\leq T. \end{equation*}$$

(ii)

$\forall t\in [0,T],\qquad \sum _{ \pi _n} (S(t_{j + 1}\wedge t) - S(t_j\wedge t))^{\otimes p} \quad \mathop{\to }^{n\to \infty } \quad [S]^p(t).$

Proof.

Let $v \in \mathbb{R}^d$. Before passing to the limit, the function

$$\begin{equation*} \sum _{\substack{[t_j,t_{j+1}] \in \pi _n: \\ t_{j} \le t}} \langle v^{\otimes p}, (S(t_{j + 1}) - S(t_j))^{\otimes p} \rangle = \sum _{\substack{[t_j,t_{j+1}] \in \pi _n: \\ t_{j} \le t}} | v \cdot (S(t_{j + 1}) - S(t_j))|^{p} \end{equation*}$$

is increasing in $t$, and therefore it defines a finite (positive) measure. By assumption, this measure converges weakly to the measure defined by $(a,b]\!\mapsto \!\int _0^T\langle \mathbf{1}_{(a,b]} v^{\otimes p}, \mathrm{d}\mu _S\rangle$. In particular, we have

$$\begin{equation*} \langle v^{\otimes p}, [S]^p(t+h) - [S]^p(t)\rangle = \int _{0}^T \langle \mathbf{1}_{(t,t+h]} v^{\otimes p}, \mathrm{d}\mu _S\rangle \ge 0. \end{equation*}$$

Thus, $\langle v^{\otimes p}, [S]^p\rangle$ is increasing for all $v \in \mathbb{R}^d$, and from here it is easy to see that $[S]^p$ has finite variation (apply, e.g., polarization to go from $v^{\otimes p}$ to $v_1 \otimes \dots \otimes v_p$).

■

Theorem 4.3 (Change of variable formula for paths with finite $p$th variation).

Let $p \in \mathbb{N}$ be even, let $(\pi _n)$ be a sequence of partitions of $[0,T]$, and let $S \in V_p(\pi )\cap C([0,T],\mathbb{R}^d)$. Then for all $f \in C^p(\mathbb{R}^d,\mathbb{R})$ the limit of compensated Riemann sums

$$\begin{equation*} \int _0^t \langle \nabla f(S(s)) , \mathrm{d}S(s) \rangle \!:=\! \lim _{n \rightarrow \infty } \sum _{[t_j, t_{j + 1}] \in \pi _n} \sum _{k=1}^{p-1} \frac{1}{k!} \langle \nabla ^k f(S(t_j)), (S(t_{j + 1} \wedge t) - S(t_j \wedge t))^{\otimes k} \rangle \end{equation*}$$

exists for every $t\in [0,T]$ and satisfies the pathwise change of variable formula:

$$\begin{equation*} f(S(t)) - f(S(0)) = \int _0^t \langle \nabla f(S(s)) , \mathrm{d}S(s) \rangle + \frac{1}{p!} \int _0^t \langle \nabla ^p f(S(s)), \mathrm{d}[S]^p(s) \rangle . \end{equation*}$$

Proof.

The proof follows similar ideas to the case $p=2$. By applying a Taylor expansion at order $p$ to the increments of $f(S)$ along the partition, we obtain

$$\begin{equation} \begin{aligned} f (S(t)) - f (S(0)) &= \sum _{[t_j, t_{j + 1}] \in \pi _n} (f (S(t_{j + 1}\wedge t)) - f (S(t_j \wedge t)))\\ &= \sum _{[t_j, t_{j + 1}] \in \pi _n} \sum _{k=1}^p \frac{1}{k!} \langle \nabla ^{k}f (S(t_j)), (S(t_{j + 1}\wedge t) - S(t_j \wedge t))^{\otimes k} \rangle \\ &\qquad {} + \sum _{[t_j, t_{j + 1}] \in \pi _n} \int _0^1 \mathrm{d}\lambda \frac{(1 - \lambda )^{p - 1}}{(p - 1) !} \times \begin{aligned}[t] \big \langle \big (\nabla ^{p} f(S(t_j) + \lambda (S(t_{j + 1} \wedge t) - S(t_j \wedge t))) - \nabla ^{p} f (S(t_j))\big ), \\ (S(t_{j + 1} \wedge t) - S(t_j \wedge t))^{\otimes p} \big \rangle . \end{aligned} \end{aligned} {\tag{9}} \end{equation}$$

As in the proof of Theorem 1.5 we assume that $f$ is compactly supported and use this to show that the remainder on the right hand side vanishes as $n \rightarrow \infty$. Since $S \in V_p(\pi )$ we know that

$$\begin{align*} &\lim _{n\to \infty } \sum _{[t_j, t_{j + 1}] \in \pi _n} \frac{1}{p!} \langle \nabla ^k f (S(t_j)), (S(t_{j + 1}\wedge t) - S(t_j \wedge t))^{\otimes p} \rangle \\ &\qquad = \frac{1}{p!} \int _0^t \langle \nabla ^p f(S(s)), \mathrm{d}[S]^p(s)\rangle , \end{align*}$$

and therefore we obtain from 9

$$\begin{align*} \lefteqn{\lim _{n \to \infty } \sum _{[t_j, t_{j + 1}] \in \pi _n} \sum _{k=1}^{p-1} \frac{1}{k!} \langle \nabla ^{k}f (S(t_j)), (S(t_{j + 1}\wedge t) - S(t_j \wedge t))^{\otimes k} \rangle }\qquad \\ &= f(S(t)) - f(S(0)) - \frac{1}{p!} \int _0^t f^{(p)}(S(s)) \mathrm{d}[S]^p(s), \end{align*}$$

and we simply define $\int _0^t \langle \nabla f(S(s)), \mathrm{d}S(s)\rangle$ as the limit on the left hand side.

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4.2. Relation with rough path integration

To explain the link between Föllmer’s pathwise Itô integral and rough path integration 21, Friz and Hairer 17, Chapter 5.3 introduced the notion of (second order) reduced rough path.

Definition 4.4.

Let $\alpha \in (1/3,1/2)$. We set $\Delta _T :=\{(s,t): 0 \le s \le t \le T\}$. A reduced rough path of regularity $\alpha$ is a pair $(X, \mathbb{X})\colon \Delta _T \to \mathbb{R}^d \oplus \mathbb{\operatorname {Sym}}_2(\mathbb{R}^d)$, such that

(i)

there exists $C>0$ with$$\begin{equation*} |X_{s,t}| + \sqrt {|\mathbb{X}_{s,t}|} \le C |t-s|^\alpha , \qquad (s,t) \in \Delta _T; \end{equation*}$$

(ii)

the reduced Chen relation holds$$\begin{equation*} \mathbb{X}_{s,t} - \mathbb{X}_{s,u} - \mathbb{X}_{u,t} = \operatorname {Sym}(X_{s,u} \otimes X_{u,t}), \qquad (s,u), (u,t) \in \Delta _T, \end{equation*}$$

where $\operatorname {Sym}(\cdot )$ denotes the symmetric part.

Friz and Hairer 17 also show that, for any $S \in V_2(\pi )$, there is a canonical candidate for a reduced rough path. Indeed, the pair

$$\begin{equation*} X_{s,t} :=S(t) - S(s),\qquad \mathbb{X}_{s,t} :=\frac{1}{2} X_{s,t} \otimes X_{s,t} - \frac{1}{2} ([S]^2(t) - [S]^2(s)), \end{equation*}$$

satisfies the reduced Chen relation. But in general we do not know anything about the Hölder regularity of $S \in V_2(\pi )$, because for any continuous path $S$ there exists a sequence of partitions $(\pi _n)$ with $S \in V_2(\pi )$ and $[S]^2 \equiv 0$; see 16. If, however, we take the dyadic Lebesgue partition $(\pi _n)$ generated by $S$ as in Definition 3.3 and if $S \in V_2(\pi )$, then it follows from 4, Lemme 1⁠¹ that $S$ has finite $q$-variation for any $q>2$. So in that case every $S \in V_2(\pi )$ corresponds to a reduced rough path with $p$-variation regularity. Rather than adapting Definition 4.4 from Hölder to $p$-variation regularity, we directly introduce a concept of higher-order reduced rough paths. For that purpose we first define the concept of control function.

¹

Note that for $\lambda >0$ the path $S$ has finite $q$-variation if and only if $\lambda ^{-1} S$ has finite $q$-variation, and therefore we can assume that $\lambda = 1$ in 4, Lemme 1.

✖

Definition 4.5.

A control function is a continuous map $c \colon \Delta _T \to \mathbb{R}_+$ such that $c(t,t)=0$ for all $t \in [0,T]$ and such that $c(s,u) + c(u,t) \le c(s,t)$ for all $0 \le s \le u \le t \le T$.

A function $f \colon [0,T] \to \mathbb{R}^d$ has finite $p$-variation if and only if there exists a control function $c$ with $|f(t) - f(s)|^p \le c(s,t)$, and in that case $\| f \|_{\operatorname {p-var}} \le c(0,T)^{1/p}$.

Definition 4.6.

Let $p \ge 1$. A reduced rough path of finite $p$-variation is a tuple

$$\begin{equation*} \mathbb{X} = (1,\mathbb{X}^1, \dots , \mathbb{X}^{\lfloor p \rfloor })\colon \Delta _T \longrightarrow \mathbb{S}_{\lfloor p\rfloor }(\mathbb{R}^d), \end{equation*}$$

such that

(i)

there exists a control function $c$ with$$\begin{equation*} \sum _{k=1}^{\lfloor p\rfloor } |\mathbb{X}^k_{s,t}|^{p/k} \le c(s,t), \qquad (s,t) \in \Delta _T; \end{equation*}$$

(ii)

the reduced Chen relation holds$$\begin{equation*} \mathbb{X}_{s,t} = \operatorname {Sym}(\mathbb{X}_{s,u} \otimes \mathbb{X}_{u,t}), \qquad (s,u), (u,t) \in \Delta _T, \end{equation*}$$

where the symmetric part of $T \in T_k(\mathbb{R}^d)$ is defined as$$\begin{equation*} \operatorname {Sym}(T) :=\frac{1}{k!} \sum _{\sigma \in \mathfrak{S}_k} \sigma T,\qquad \sigma T(v_1, \dots , v_k) :=T(v_{\sigma 1},\dots , v_{\sigma k}), \end{equation*}$$

where the sum is across the group of permutations $\mathfrak{S}_k$ of $\{1, \dots , k\}$.

Lemma 4.7.

Let $S \in C([0,T],\mathbb{R}^d)$ and let $(\pi _n)$ be the dyadic Lebesgue partition generated by $S$. Let $p \ge 1$ and assume that $S \in V_p(\pi )$. Then for any $q>p$ with $\lfloor q \rfloor = \lfloor p \rfloor$ we obtain a reduced rough path of finite $q$-variation by setting $\mathbb{X}^0_{s,t} :=1$,

$$\begin{gather*} \mathbb{X}^k_{s,t} :=\frac{1}{k!} (S(t) - S(s))^{\otimes k},\qquad k=1,\dots , \lfloor p \rfloor - 1,\\ \mathbb{X}^{\lfloor p \rfloor }_{s,t} :=\frac{1}{\lfloor p \rfloor !} (S(t) - S(s))^{\otimes \lfloor p \rfloor } - \frac{1}{\lfloor p \rfloor !} ([S]^p(t) - [S]^p(s)). \end{gather*}$$

Proof.

Let $q>p$. As discussed above we know that $S$ has finite $q$-variation, so let us start by setting

$$\begin{equation*} \tilde{c}(s,t) :=\|S \|_{\operatorname {q-var},[s,t]}^q :=\sup _{\pi \in \Pi ([s,t])} \sum _{[t_j, t_{j + 1}] \in \pi } |S(t_{j + 1}) - S(t_j)|^q,\qquad (s,t) \in \Delta _T, \end{equation*}$$

which is a control function such that

$$\begin{equation*} \sum _{k=1}^{\lfloor p\rfloor } |\mathbb{X}^k_{s,t}|^{q/k} \le C_{d,p} \big ( \tilde{c}(s,t) + |[S]^p(t) - [S]^p(s)|^{q/\lfloor p \rfloor } \big ), \end{equation*}$$

with a constant $C_{d,p} >0$ that only depends on the dimension $d$ and on $p$. By Property 4.2 the path $[S]^p$ has finite variation and therefore it also has finite $q/\lfloor p \rfloor$-variation, so

$$\begin{equation*} \tilde{\tilde{c}}(s,t) :=\| [S]^p \|_{\operatorname {q/\lfloor p \rfloor -var},[s,t]}^{q/\lfloor p \rfloor } \end{equation*}$$

defines another control function. Therefore, $c(s,t) :=C_{d,p} ( \tilde{c}(s,t) + \tilde{\tilde{c}}(s,t))$ is a control function for which the analytic property (i) in Definition (i) holds.

To show the reduced Chen relation let us write $\mathfrak{S}_{\ell , k}$ for $0 \le \ell , k$ for the shuffles of words of length $\ell ,k$, i.e., for those permutations $\sigma \in \mathfrak{S}_{\ell + k}$ which satisfy $\sigma i < \sigma j$ for all $1 \le i < j \le \ell$, respectively, $\ell +1 \le i < j \le k$. Note that there are $\binom{\ell +k}{\ell }$ shuffles in $\mathfrak{S}_{\ell ,k}$. We have for $k < \lfloor p \rfloor$

$$\begin{align*} \mathbb{X}_{s,t}^k & = \frac{1}{k!} (S(t) - S(s))^{\otimes k} = \frac{1}{k!} (S(t) - S(u) + S(u) - S(s))^{\otimes k} \\ & = \frac{1}{k!} \sum _{\ell = 0}^k \sum _{\sigma \in \mathfrak{S}_{\ell , k-\ell }} \sigma \big ((S(u) - S(s))^{\otimes \ell } \otimes (S(t) - S(u))^{\otimes (k-\ell )}\big ), \end{align*}$$

where we set $v^{\otimes 0} :=1$ for all $v \in R^d$. On the other hand, if $\mathcal{P}_k$ denotes the projection onto $T_k(\mathbb{R}^d)$, then for $k < \lfloor p \rfloor$

$$\begin{align*} &\mathcal{P}_k (\operatorname {Sym}(\mathbb{X}_{s,u} \otimes \mathbb{X}_{u,t}))\\ &\qquad \begin{aligned}[t] & = \sum _{\ell =0}^k \operatorname {Sym}( \mathbb{X}^\ell _{s,u} \otimes \mathbb{X}^{k-\ell }_{u,t}) \\ & = \sum _{\ell =0}^k \frac{1}{\ell ! (k-\ell )!} \operatorname {Sym}\big ((S(u) - S(s))^{\otimes \ell } \otimes (S(t) - S(u))^{\otimes (k-\ell )}\big ) \\ & = \sum _{\ell =0}^k \frac{1}{\ell ! (k-\ell )!} \binom{k}{\ell }^{-1} \sum _{\sigma \in \mathfrak{S}_{\ell , k - \ell }} \sigma ((S(u) - S(s))^{\otimes \ell } \otimes (S(t) - S(u))^{\otimes (k-\ell )}\big ) \\ & = \mathbb{X}^k_{s,t}, \end{aligned} \end{align*}$$

which proves the reduced Chen relation for $k < \lfloor p \rfloor$. For $k = \lfloor p \rfloor$ we get the same relation by noting that $[S]^p$ is already symmetric and therefore

$$\begin{equation*} \operatorname {Sym}([S]^p(t) - [S]^p(s)) = [S]^p(t) - [S]^p(s). \end{equation*}$$ ■

The following space of (higher-order) controlled paths in the sense of Gubinelli 19 is defined for example in 17, Chapter 4.5. We adapt the definition to paths that are controlled in the $p$-variation sense by a reduced rough path. If $\ell < k$ and $T \in T_\ell$, $\tilde{T} \in T_k$, then we interpret

$$\begin{equation*} \langle T, \tilde{T} \rangle \in T_{k - \ell }, \qquad \langle T, \tilde{T} \rangle (v_1, \dots , v_{k-\ell }) :=\langle T \otimes (v_1 \otimes \dots \otimes v_{k - \ell }), \tilde{T} \rangle , \end{equation*}$$

and similarly for $\langle \tilde{T}, T\rangle$.

Definition 4.8.

Let $p \ge 1$ and let $\mathbb{X}$ be a reduced rough path of finite $p$-variation. A path

$$\begin{equation*} Y = (Y^1, \dots , Y^{\lfloor p \rfloor }) \in C([0,T],\mathbb{S}_{\lfloor p\rfloor }(\mathbb{R}^d)) \end{equation*}$$

is controlled by $\mathbb{X}$ if there exists a control function $c$ such that

$$\begin{equation*} \sum _{\ell = 1}^{\lfloor p \rfloor } \Big | Y^\ell (t) - \sum _{k=\ell }^{\lfloor p \rfloor } \langle Y^k(s), \mathbb{X}^{k-\ell }_{s,t} \rangle \Big |^{\frac{p}{\lfloor p \rfloor - \ell + 1}} \le c(s,t),\qquad (s,t) \in \Delta _T. \end{equation*}$$

In that case we write $Y \in \mathcal{D}^{\lfloor p \rfloor /p}_{\mathbb{X}}([0,T])$.

Example 4.9.

Let $p \ge 1$, let $S$, $\mathbb{X}$ and $q$ be as in Lemma 4.7, and let $f \in C^{\lfloor q\rfloor }(\mathbb{R}^d, \mathbb{R})$. Then $Y^0 :=1$,

$$\begin{equation*} Y^k(s) :=\nabla ^{k} f(S(s)),\qquad k=1,\dots , \lfloor q \rfloor , \end{equation*}$$

defines a controlled path in $\mathcal{D}^{\lfloor q \rfloor /q}_{\mathbb{X}}([0,T])$. Indeed, as we discussed above $\nabla ^k f(S(s)) \in \operatorname {Sym}_k(\mathbb{R}^d)$ for all $k = 1,\dots , \lfloor q \rfloor$, and by Taylor’s formula we have for $\ell \in \{1, \dots , \lfloor q \rfloor \}$

$$\begin{align*} Y^\ell (t) & = \nabla ^\ell f(S(t))\\ & = \sum _{k=\ell }^{\lfloor q \rfloor } \frac{1}{(k-\ell )!} \langle \nabla ^{k} f(S(s)), (S(t) - S(s))^{\otimes (k-\ell )} \rangle + O(c(s,t)^{(\lfloor q \rfloor - \ell + 1)/q}) \\ & = \sum _{k=\ell }^{\lfloor q \rfloor } \langle Y^k(s), \mathbb{X}_{s,t}^{k-\ell } \rangle + O(c(s,t)^{(\lfloor q \rfloor - \ell + 1)/q}). \end{align*}$$

Proposition 4.10.

Let $p \ge 1$, let $\mathbb{X}$ be a reduced rough path of finite $p$-variation, and let $Y \in \mathcal{D}^{\lfloor p \rfloor /p}_{\mathbb{X}}([0,T])$. Then the rough path integral

$$\begin{equation*} I_{\mathbb{X}}(Y)(t)=\int _0^t \langle Y(s), \mathrm{d}\mathbb{X}(s)\rangle = \lim _{\substack{\pi \in \Pi ([0,t])\\ |\pi | \to 0}} \sum _{[t_j,t_{j+1}] \in \pi } \sum _{k=1}^{\lfloor p \rfloor } \langle Y^k(t_j), \mathbb{X}^k_{t_j,t_{j+1}}\rangle ,\qquad t \!\in \! [0,T], \end{equation*}$$

defines a function in $C([0,T],\mathbb{R})$, and it is the unique function with $I_{\mathbb{X}}(Y)(0)=0$ for which there exists a control function $c$ with

$$\begin{equation*} \Big |\int _s^t \langle Y(r), \mathrm{d}\mathbb{X}(r)\rangle - \sum _{k=1}^{\lfloor p \rfloor } \langle Y^k(s), \mathbb{X}^k_{s,t}\rangle \Big | \lesssim c(s,t)^{\frac{\lfloor p \rfloor +1}{p}},\qquad (s,t) \in \Delta _T. \end{equation*}$$

Proof.

This follows from classical arguments (Theorem 4.3 in 22; see also 19) once we show that for $0 \le s \le u \le t \le T$

$$\begin{equation*} \sum _{k=1}^{\lfloor p \rfloor } \langle Y^k(s), \mathbb{X}^k_{s,t}\rangle - \sum _{k=1}^{\lfloor p \rfloor } \langle Y^k(s), \mathbb{X}^k_{s,u}\rangle - \sum _{k=1}^{\lfloor p \rfloor } \langle Y^k(u), \mathbb{X}^k_{u,t}\rangle = O(c(s,t)^{\frac{\lfloor p \rfloor +1}{p}}), \end{equation*}$$

where $c$ is a control function such that the estimates in Definitions 4.6 and 4.8 hold. But

$$\begin{align*} \sum _{k=1}^{\lfloor p \rfloor } \langle Y^k(u), \mathbb{X}^k_{u,t}\rangle & = \sum _{k=1}^{\lfloor p \rfloor } \Big (\sum _{\ell =k}^{\lfloor p \rfloor } \langle Y^\ell (s), \mathbb{X}^{\ell -k}_{s,u} \otimes \mathbb{X}^k_{u,t}\rangle + O\big (c(s,u)^{\frac{\lfloor p \rfloor - k + 1}{p}} c(u,t)^{\frac{k}{p}}\big ) \Big ) \\ & = \sum _{k=1}^{\lfloor p \rfloor } \sum _{\ell =1}^k \langle Y^k(s), \mathbb{X}^{k-\ell }_{s,u} \otimes \mathbb{X}^\ell _{u,t}\rangle + O\big (c(s,t)^{\frac{\lfloor p\rfloor +1}{p}}\big ) \\ & = \sum _{k=1}^{\lfloor p \rfloor } \langle Y^k(s), \mathcal{P}_k(\operatorname {Sym}(\mathbb{X}_{s,u} \otimes \mathbb{X}_{u,t})) - \mathbb{X}^k_{s,u}\rangle + O\big (c(s,t)^{\frac{\lfloor p\rfloor +1}{p}}\big ), \end{align*}$$

where in the last step we used that $Y^k(s)$ is symmetric. Therefore, the reduced Chen relation gives

$$\begin{align*} &\sum _{k=1}^{\lfloor p \rfloor } \langle Y^k(s), \mathbb{X}^k_{s,t}\rangle - \sum _{k=1}^{\lfloor p \rfloor } \langle Y^k(s), \mathbb{X}^k_{s,u}\rangle - \sum _{k=1}^{\lfloor p \rfloor } \langle Y^k(u), \mathbb{X}^k_{u,t}\rangle \\ &\qquad \begin{aligned}[t] & = \sum _{k=1}^{\lfloor p \rfloor } \langle Y^k(s), \mathbb{X}^k_{s,t} - \mathbb{X}^k_{s,u} - \mathcal{P}_k(\operatorname {Sym}(\mathbb{X}_{s,u} \otimes \mathbb{X}_{u,t})) + \mathbb{X}^k_{s,u}\rangle + O\big (c(s,t)^{\frac{\lfloor p\rfloor +1}{p}}\big ) \\ & = \sum _{k=1}^{\lfloor p \rfloor } \langle Y^k(s), \mathbb{X}^k_{s,t} - \mathbb{X}^k_{s,t}\rangle + O\big (c(s,t)^{\frac{\lfloor p\rfloor +1}{p}}\big ) = O\big (c(s,t)^{\frac{\lfloor p\rfloor +1}{p}}\big ), \end{aligned} \end{align*}$$

which concludes the proof.

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Corollary 4.11.

Let $p\in \mathbb{N}$ be an even integer and let $q, S,\mathbb{X}, f$ be as in Example 4.9. Then

$$\begin{equation*} \int _0^t \langle \nabla f(S(s)), \mathrm{d}\mathbb{X}(s)\rangle = \int _0^t \langle \nabla f(S(s)), \mathrm{d}S(s) \rangle ,\qquad t \in [0,T], \end{equation*}$$

where the left hand side denotes the rough path integral of Proposition 4.10 and the right hand side is the integral of Theorem 4.3.

Proof.

It suffices to show that

$$\begin{equation*} \int _0^t \langle \nabla f(S(s)), \mathrm{d}\mathbb{X}(s)\rangle = f(S(t)) - f(S(0)) - \frac{1}{p!} \int _0^t \langle \nabla ^p f(S(s)), \mathrm{d}[S]^p(s) \rangle , \end{equation*}$$

and since

$$\begin{equation*} \lim _{\substack{\pi \in \Pi ([0,t])\\ |\pi | \to 0}} \sum _{[t_j,t_{j+1}] \in \pi } \langle \nabla ^p f(S(t_j)), [S]^p(t_{j+1}) - [S]^p(t_j)\rangle = \int _0^t \langle \nabla ^p f(S(s)), \mathrm{d}[S]^p(s) \rangle , \end{equation*}$$

this is equivalent to

$$\begin{equation*} \lim _{\substack{\pi \in \Pi ([0,t])\\ |\pi | \to 0}} \sum _{[t_j,t_{j+1}] \in \pi } \sum _{k=1}^{ p } \langle \nabla ^k f(S(t_j)), \frac{1}{k!}(S(t_{j+1}) - S(t_j))^{\otimes k} \rangle = f(S(t)) - f(S(0)). \end{equation*}$$

The last identity can be shown by writing $f(S(t)) - f(S(0))$ as a telescoping sum and by performing a Taylor expansion up to order $p$ and controlling the remainder term as in the proof of Theorem 4.3.

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