In stochastic analysis, a rough path is a generalization of the notion of smooth path allowing to construct a robust solution theory for controlled differential equations driven by classically irregular signals, for example a Wiener process. The theory was developed in the 1990s by Terry Lyons.^[1][2][3] Several accounts of the theory are available.^[4][5][6][7]

Rough path theory is focused on capturing and making precise the interactions between highly oscillatory and non-linear systems. It builds upon the harmonic analysis of L.C. Young, the geometric algebra of K.T. Chen, the Lipschitz function theory of H. Whitney and core ideas of stochastic analysis. The concepts and the uniform estimates have widespread application in pure and applied Mathematics and beyond. It provides a toolbox to recover with relative ease many classical results in stochastic analysis (Wong-Zakai, Stroock-Varadhan support theorem, construction of stochastic flows, etc) without using specific probabilistic properties such as the martingale property or predictability. The theory also extends Itô's theory of SDEs far beyond the semimartingale setting. At the heart of the mathematics is the challenge of describing a smooth but potentially highly oscillatory and multidimensional path ${\displaystyle x_{t}}$ effectively so as to accurately predict its effect on a nonlinear dynamical system ${\displaystyle \mathrm {d} y_{t}=f(y_{t})\,\mathrm {d} x_{t},y_{0}=a}$. The Signature is a homomorphism from the monoid of paths (under concatenation) into the grouplike elements of the free tensor algebra. It provides a graduated summary of the path ${\displaystyle x}$. This noncommutative transform is faithful for paths up to appropriate null modifications. These graduated summaries or features of a path are at the heart of the definition of a rough path; locally they remove the need to look at the fine structure of the path. Taylor's theorem explains how any smooth function can, locally, be expressed as a linear combination of certain special functions (monomials based at that point). Coordinate iterated integrals (terms of the signature) form a more subtle algebra of features that can describe a stream or path in an analogous way; they allow a definition of rough path and form a natural linear "basis" for continuous functions on paths.

Martin Hairer used rough paths to construct a robust solution theory for the KPZ equation.^[8] He then proposed a generalization known as the theory of regularity structures^[9] for which he was awarded a Fields medal in 2014.

Motivation

Rough path theory aims to make sense of the controlled differential equation

${\displaystyle \mathrm {d} Y_{t}^{i}=\sum _{j=1}^{d}V_{j}^{i}(Y_{t})\,\mathrm {d} X_{t}^{j}.}$

where the control, the continuous path ${\displaystyle X_{t}}$ taking values in a Banach space, need not be differentiable nor of bounded variation. A prevalent example of the controlled path ${\displaystyle X_{t}}$ is the sample path of a Wiener process. In this case, the aforementioned controlled differential equation can be interpreted as a stochastic differential equation and integration against "${\displaystyle \mathrm {d} X_{t}^{j}}$" can be defined in the sense of Itô. However, Itô's calculus is defined in the sense of ${\displaystyle L^{2}}$ and is in particular not a pathwise definition. Rough paths give an almost sure pathwise definition of stochastic differential equations. The rough path notion of solution is well-posed in the sense that if ${\displaystyle X(n)_{t}}$ is a sequence of smooth paths converging to ${\displaystyle X_{t}}$ in the ${\displaystyle p}$-variation metric (described below), and

${\displaystyle \mathrm {d} Y(n)_{t}^{i}=\sum _{j=1}^{d}V_{j}^{i}(Y_{t})\,\mathrm {d} X(n)_{t}^{j};}$

${\displaystyle \mathrm {d} Y_{t}^{i}=\sum _{j=1}^{d}V_{j}^{i}(Y_{t})\,\mathrm {d} X_{t}^{j},}$

then ${\displaystyle Y(n)}$ converges to ${\displaystyle Y}$ in the ${\displaystyle p}$-variation metric. This continuity property and the deterministic nature of solutions makes it possible to simplify and strengthen many results in Stochastic Analysis, such as the Freidlin-Wentzell's Large Deviation theory^[10] as well as results about stochastic flows.

In fact, rough path theory can go far beyond the scope of Itô and Stratonovich calculus and allows to make sense of differential equations driven by non-semimartingale paths, such as Gaussian processes and Markov processes.^[11]

Definition of a rough path

Rough paths are paths taking values in the truncated free tensor algebra (more precisely: in the free nilpotent group embedded in the free tensor algebra), which this section now briefly recalls. The tensor powers of ${\displaystyle \mathbb {R} ^{d}}$, denoted ${\displaystyle {\big (}\mathbb {R} ^{d}{\big )}^{\otimes n}}$, are equipped with the projective norm ${\displaystyle \Vert \cdot \Vert }$ (see Topological tensor product, note that rough path theory in fact works for a more general class of norms). Let ${\displaystyle T^{(n)}(\mathbb {R} ^{d})}$ be the truncated tensor algebra

${\displaystyle T^{(n)}(\mathbb {R} ^{d})=\bigoplus _{i=0}^{n}{\big (}\mathbb {R} ^{d}{\big )}^{\otimes i},}$ where by convention ${\displaystyle (\mathbb {R} ^{d})^{\otimes 0}\cong \mathbb {R} }$.

Let ${\displaystyle \triangle _{0,1}}$ be the simplex ${\displaystyle \{(s,t):0\leq s\leq t\leq 1\}}$. Let ${\displaystyle p\geq 1}$. Let ${\displaystyle \mathbf {X} }$ and ${\displaystyle \mathbf {Y} }$ be continuous maps ${\displaystyle \triangle _{0,1}\to T^{(\lfloor p\rfloor )}(\mathbb {R} ^{d})}$. Let ${\displaystyle \mathbf {X} ^{j}}$ denote the projection of ${\displaystyle \mathbf {X} }$ onto ${\displaystyle j}$-tensors and likewise for ${\displaystyle \mathbf {Y} ^{j}}$. The ${\displaystyle p}$-variation metric is defined as

${\displaystyle d_{p}\left(\mathbf {X} ,\mathbf {Y} \right):=\max _{j=1,\ldots ,\lfloor p\rfloor }\sup _{0=t_{0}<t_{1}<\cdots <t_{n}=1}\left(\sum _{i=0}^{n-1}\Vert \mathbf {X} _{t_{i},t_{i+1}}^{j}-\mathbf {Y} _{t_{i},t_{i+1}}^{j}\Vert ^{\frac {p}{j}}\right)^{\frac {j}{p}}}$

where the supremum is taken over all finite partitions ${\displaystyle \{0=t_{0}<t_{1}<\cdots <t_{n}=1\}}$ of ${\displaystyle [0,1]}$.

A continuous function ${\displaystyle \mathbf {X}$ :\triangle _{0,1}\rightarrow T^{(\lfloor p\rfloor )}(\mathbb {R} ^{d})} is a ${\displaystyle p}$-geometric rough path if there exists a sequence of paths with finite total variation ${\displaystyle X(1),X(2),\ldots }$ such that

${\displaystyle \mathbf {X} (n)_{s,t}=\left(1,\int _{s<s_{1}<t}\mathrm {d} X(n)_{s_{1}},\ldots ,\int _{s<s_{1}<\cdots <s_{\lfloor p\rfloor }<t}\,\mathrm {d} X(n)_{s_{1}}\otimes \cdots \otimes \mathrm {d} X(n)_{s_{\lfloor p\rfloor }}\right)}$

converges in the ${\displaystyle p}$-variation metric to ${\displaystyle \mathbf {X} }$ as ${\displaystyle n\rightarrow \infty }$.^[12]

Universal limit theorem

A central result in rough path theory is Lyons' Universal Limit theorem.^[1] One (weak) version of the result is the following: Let ${\displaystyle X(n)}$ be a sequence of paths with finite total variation and let

${\displaystyle \mathbf {X} (n)_{s,t}=\left(1,\int _{s<s_{1}<t}\mathrm {d} X(n)_{s_{1}},\ldots ,\int _{s<s_{1}<\ldots <s_{\lfloor p\rfloor }<t}\mathrm {d} X(n)_{s_{1}}\otimes \cdots \otimes \mathrm {d} X(n)_{s_{\lfloor p\rfloor }}\right)}$ denote the rough path lift of ${\displaystyle X(n)}$.

Suppose that ${\displaystyle \mathbf {X} (n)}$ converges in the ${\displaystyle p}$-variation metric to a ${\displaystyle p}$-geometric rough path ${\displaystyle \mathbf {X} }$ as ${\displaystyle n\to \infty }$. Let ${\displaystyle (V_{j}^{i})_{j=1,\ldots ,d}^{i=1,\ldots ,n}}$ be functions that have at least ${\displaystyle \lfloor p\rfloor }$ bounded derivatives and the ${\displaystyle \lfloor p\rfloor }$-th derivatives are ${\displaystyle \alpha }$-Hölder continuous for some ${\displaystyle \alpha >p-\lfloor p\rfloor }$. Let ${\displaystyle Y(n)}$ be the solution to the differential equation

${\displaystyle \mathrm {d} Y(n)_{t}^{i}=\sum _{j=1}^{d}V_{j}^{i}(Y(n)_{t})\,\mathrm {d} X(n)_{t}^{j}}$

and let ${\displaystyle \mathbf {Y} (n)}$ be defined as

${\displaystyle \mathbf {Y} (n)_{s,t}=\left(1,\int _{s<s_{1}<t}\,\mathrm {d} Y(n)_{s_{1}},\ldots ,\int _{s<s_{1}<\ldots <s_{\lfloor p\rfloor }<t}\mathrm {d} Y(n)_{s_{1}}\otimes \cdots \otimes \mathrm {d} Y(n)_{s_{\lfloor p\rfloor }}\right).}$

Then ${\displaystyle \mathbf {Y} (n)}$ converges in the ${\displaystyle p}$-variation metric to a ${\displaystyle p}$-geometric rough path ${\displaystyle \mathbf {Y} }$.

Moreover, ${\displaystyle \mathbf {Y} }$ is the solution to the differential equation

${\displaystyle \mathrm {d} Y_{t}^{i}=\sum _{j=1}^{d}V_{j}^{i}(Y_{t})\,\mathrm {d} X_{t}^{j}\qquad (\star )}$

driven by the geometric rough path ${\displaystyle \mathbf {X} }$.

Concisely, the theorem can be interpreted as saying that the solution map (aka the Itô-Lyons map) ${\displaystyle \Phi :G\Omega _{p}(\mathbb {R} ^{d})\to G\Omega _{p}(\mathbb {R} ^{e})}$ of the RDE ${\displaystyle (\star )}$ is continuous (and in fact locally lipschitz) in the ${\displaystyle p}$-variation topology. Hence rough paths theory demonstrates that by viewing driving signals as rough paths, one has a robust solution theory for classical stochastic differential equations and beyond.

Examples of rough paths

Applications in stochastic analysis

Controlled rough path

Controlled rough paths, introduced by M. Gubinelli,^[5] are paths ${\displaystyle \mathbf {Y} }$ for which the rough integral

${\displaystyle \int _{s}^{t}\mathbf {Y} _{u}\,\mathrm {d} X_{u}}$

can be defined for a given geometric rough path ${\displaystyle X}$.

More precisely, let ${\displaystyle L(V,W)}$ denote the space of bounded linear maps from a Banach space ${\displaystyle V}$ to another Banach space ${\displaystyle W}$.

Given a ${\displaystyle p}$-geometric rough path

${\displaystyle \mathbf {X} =(1,\mathbf {X} ^{1},\ldots ,\mathbf {X} ^{\lfloor p\rfloor })}$

on ${\displaystyle \mathbb {R} ^{d}}$, a ${\displaystyle \gamma }$-controlled path is a function ${\displaystyle \mathbf {Y} _{s}=(\mathbf {Y} _{s}^{0},\mathbf {Y} _{s}^{1},\ldots ,\mathbf {Y} _{s}^{\lfloor \gamma \rfloor })}$ such that ${\displaystyle \mathbf {Y} ^{j}:[0,1]\rightarrow L((\mathbb {R} ^{d})^{\otimes j+1},\mathbb {R} ^{n})}$ and that there exists ${\displaystyle M>0}$ such that for all ${\displaystyle 0\leq s\leq t\leq 1}$ and ${\displaystyle j=0,1,\ldots ,\lfloor \gamma \rfloor }$,

${\displaystyle \Vert \mathbf {Y} _{s}^{j}\Vert \leq M}$

and

${\displaystyle \left\|\mathbf {Y} _{t}^{j}-\sum _{i=0}^{\lfloor \gamma \rfloor -j}\mathbf {Y} _{s}^{j+i}\mathbf {X} _{s,t}^{i}\right\|\leq M|t-s|^{\frac {\gamma -j}{p}}.}$

Signature

Let ${\displaystyle X:[0,1]\rightarrow \mathbb {R} ^{d}}$ be a continuous function with finite total variation. Define

${\displaystyle S(X)_{s,t}=\left(1,\int _{s<s_{1}<t}\mathrm {d} X_{s_{1}},\int _{s<s_{1}<s_{2}<t}\mathrm {d} X_{s_{1}}\otimes \mathrm {d} X_{s_{2}},\ldots ,\int _{s<s_{1}<\cdots <s_{n}<t}\mathrm {d} X_{s_{1}}\otimes \cdots \otimes \mathrm {d} X_{s_{n}},\ldots \right).}$

The signature of a path is defined to be ${\displaystyle S(X)_{0,1}}$.

The signature can also be defined for geometric rough paths. Let ${\displaystyle \mathbf {X} }$ be a geometric rough path and let ${\displaystyle \mathbf {X} (n)}$ be a sequence of paths with finite total variation such that

${\displaystyle \mathbf {X} (n)_{s,t}=\left(1,\int _{s<s_{1}<t}\,\mathrm {d} X(n)_{s_{1}},\ldots ,\int _{s<s_{1}<\cdots <s_{\lfloor p\rfloor }<t}\,\mathrm {d} X(n)_{s_{1}}\otimes \cdots \otimes \mathrm {d} X(n)_{s_{\lfloor p\rfloor }}\right).}$

converges in the ${\displaystyle p}$-variation metric to ${\displaystyle \mathbf {X} }$. Then

${\displaystyle \int _{s<s_{1}<\cdots <s_{N}<t}\,\mathrm {d} X(n)_{s_{1}}\otimes \cdots \otimes \mathrm {d} X(n)_{s_{N}}}$

converges as ${\displaystyle n\rightarrow \infty }$ for each ${\displaystyle N}$. The signature of the geometric rough path ${\displaystyle \mathbf {X} }$ can be defined as the limit of ${\displaystyle S(X(n))_{s,t}}$ as ${\displaystyle n\rightarrow \infty }$.

The signature satisfies Chen's identity,^[19] that

${\displaystyle S(\mathbf {X} )_{s,u}\otimes S(\mathbf {X} )_{u,t}=S(\mathbf {X} )_{s,t}}$

for all ${\displaystyle s\leq u\leq t}$.

Infinite dimensions

It is also possible to extend the core results in rough path theory to infinite dimensions, providing that the norm on the tensor algebra satisfies certain admissibility condition.^[24]

References