Rough path#Signature

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

:$\backslash mathrm\{d\}\; Y^i\_t\; =\; \backslash sum^d\_\{j=1\}\; V^i\_j(Y\_t)\; \backslash ,\; \backslash mathrm\{d\}X^j\_t.$

where the control, the continuous path $X\_t$ taking values in a Banach space, need not be differentiable nor of bounded variation. A prevalent example of the controlled path $X\_t$ 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 "$\backslash mathrm\{d\}X^\{j\}\_t$" can be defined in the sense of Itô. However, Itô's calculus is defined in the sense of $L^\{2\}$ 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 $X(n)\_t$ is a sequence of smooth paths converging to $X\_t$ in the $p$-variation metric (described below), and

:$\backslash mathrm\{d\}\; Y(n)^i\_t\; =\; \backslash sum^d\_\{j=1\}\; V^i\_j(Y\_t)\; \backslash ,\; \backslash mathrm\{d\}X(n)^j\_t;$

:$\backslash mathrm\{d\}\; Y^i\_t\; =\; \backslash sum^d\_\{j=1\}\; V^i\_j(Y\_t)\; \backslash ,\; \backslash mathrm\{d\}X^j\_t,$

then $Y(n)$ converges to $Y$ in the $p$-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{{cite Q|Q56689332|author1=Ledoux, Michel|author-link1=Michel Ledoux|author2=Qian, Zhongmin|author3=Zhang, Tusheng}} 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.{{cite book|last1=Friz|first1=Peter K.|last2=Victoir|first2=Nicolas|title=Multidimensional Stochastic Processes as Rough Paths: Theory and Applications|date=2010|publisher=Cambridge University Press|edition=Cambridge Studies in Advanced Mathematics}}

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 $\backslash mathbb\{R\}^\{d\}$, denoted $\backslash big(\backslash mathbb\{R\}^\{d\}\backslash big)^\{\backslash otimes\; n\}$, are equipped with the projective norm $\backslash Vert\; \backslash cdot\; \backslash Vert$ (see Topological tensor product, note that rough path theory in fact works for a more general class of norms).

Let $T^\{(n)\}(\backslash mathbb\{R\}^\{d\})$ be the truncated tensor algebra

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

Let $\backslash triangle\_\{0,1\}$ be the simplex $\backslash \{(s,t):0\backslash leq\; s\backslash leq\; t\; \backslash leq\; 1\backslash \}$.

Let $p\backslash geq\; 1$. Let $\backslash mathbf\{X\}$ and $\backslash mathbf\{Y\}$ be continuous maps $\backslash triangle\_\{0,1\}\backslash to\; T^\{(\backslash lfloor\; p\; \backslash rfloor)\}(\backslash mathbb\{R\}^\{d\})$.

Let $\backslash mathbf\{X\}^j$ denote the projection of $\backslash mathbf\{X\}$ onto $j$-tensors and likewise for $\backslash mathbf\{Y\}^\{j\}$. The $p$-variation metric is defined as

:

where the supremum is taken over all finite partitions

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

:

converges in the $p$-variation metric to $\backslash mathbf\{X\}$ as $n\backslash rightarrow\; \backslash infty$.{{Cite book | last1 = Lyons | first1 = Terry | author-link1 = Terry Lyons (mathematician)| last2 = Qian | first2 = Zhongmin| doi = 10.1093/acprof:oso/9780198506485.001.0001 | title = System Control and Rough Paths | year = 2002 | isbn = 9780198506485 | series=Oxford Mathematical Monographs| location=Oxford| publisher=Clarendon Press| zbl=1029.93001}}

A central result in rough path theory is Lyons' Universal Limit theorem. One (weak) version of the result is the following:

Let $X(n)$ be a sequence of paths with finite total variation and let

:

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

: $\backslash mathrm\{d\}\; Y(n)^i\_t\; =\; \backslash sum^d\_\{j=1\}\; V^i\_j(Y(n)\_t)\; \backslash ,\; \backslash mathrm\{d\}\; X(n)^j\_t$

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

:

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

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

: $\backslash mathrm\{d\}\; Y^i\_t\; =\; \backslash sum^d\_\{j=1\}\; V^i\_j(Y\_t)\; \backslash ,\; \backslash mathrm\{d\}\; X^j\_t\; \backslash qquad\; (\backslash star)$

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

The theorem can be interpreted as saying that the solution map (aka the Itô-Lyons map) $\backslash Phi:G\backslash Omega\_p(\backslash mathbb\{R\}^d)\backslash to\; G\backslash Omega\_p(\backslash mathbb\{R\}^e)$ of the RDE $(\backslash star)$ is continuous (and in fact locally lipschitz) in the $p$-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.

Let $(B\_t)\_\{t\backslash geq\; 0\}$ be a multidimensional standard Brownian motion. Let $\backslash circ$ denote the Stratonovich integration. Then

:

is a $p$-geometric rough path for any Stratonovich Brownian rough path.

More generally, let $B\_H(t)$ be a multidimensional fractional Brownian motion (a process whose coordinate components are independent fractional Brownian motions) with $H>\backslash frac\{1\}\{4\}$. If $B^\{m\}\_H(t)$ is the $m$-th dyadic piecewise linear interpolation of $B\_H(t)$, then

: $$

\begin{align}

\mathbf{B}^m_H(s,t) = \left(1,\int_{s

& \left. \int_{s

\end{align}

converges almost surely in the $p$-variation metric to a $p$-geometric rough path for Lyons–Victoir extension theorem.

In general, let $(X\_t)\_\{t\backslash geq0\}$ be a $\backslash mathbb\{R\}^d$-valued stochastic process. If one can construct, almost surely, functions $(s,t)\backslash rightarrow\; \backslash mathbf\{X\}^\{j\}\_\{s,t\}\; \backslash in\; \backslash big(\backslash mathbb\{R\}^d\backslash big)^\{\backslash otimes\; j\}$ so that

:$\backslash mathbf\{X\}:(s,t)\backslash rightarrow\; (1,X\_t-X\_s,\backslash mathbf\{X\}^2\_\{s,t\},\backslash ldots,\backslash mathbf\{X\}^\{\backslash lfloor\; p\; \backslash rfloor\}\_\{s,t\})$

is a $p$-geometric rough path, then $\backslash mathbf\{X\}\_\{s,t\}$ is an enhancement of the process $X$. Once an enhancement has been chosen, the machinery of rough path theory will allow one to make sense of the controlled differential equation

:$\backslash mathrm\{d\}\; Y^i\_t\; =\; \backslash sum^d\_\{j=1\}\; V^i\_j(Y\_t)\; \backslash ,\; \backslash mathrm\{d\}\; X^j\_t.$

for sufficiently regular vector fields $V^i\_j.$

Note that every stochastic process (even if it is a deterministic path) can have more than one (in fact, uncountably many) possible enhancements.{{cite journal|last1=Lyons|first1=Terry|last2=Victoir|first2=Nicholas|title=An extension theorem to rough paths|journal=Annales de l'Institut Henri Poincaré C |date=2007|doi=10.1016/j.anihpc.2006.07.004|volume=24|issue=5|pages=835–847|bibcode=2007AIHPC..24..835L|url=http://www.numdam.org/item/AIHPC_2007__24_5_835_0/|doi-access=free}} Different enhancements will give rise to different solutions to the controlled differential equations. In particular, it is possible to enhance Brownian motion to a geometric rough path in a way other than the Brownian rough path.{{cite journal|last1=Friz|first1=Peter|last2=Gassiat|first2=Paul|last3=Lyons|first3=Terry|title=Physical Brownian motion in a magnetic field as a rough path|journal=Transactions of the American Mathematical Society|date=2015|doi=10.1090/S0002-9947-2015-06272-2

|volume=367|issue=11|pages=7939–7955|arxiv=1302.2531|s2cid=59358406}} This implies that the Stratonovich calculus is not the only theory of stochastic calculus that satisfies the classical product rule

: $\backslash mathrm\{d\}(X\_t\backslash cdot\; Y\_t)\; =\; X\_t\; \backslash ,\; \backslash mathrm\{d\}\; Y\_t+Y\_t\; \backslash ,\; \backslash mathrm\{d\}\; X\_t.$

In fact any enhancement of Brownian motion as a geometric rough path will give rise a calculus that satisfies this classical product rule. Itô calculus does not come directly from enhancing Brownian motion as a geometric rough path, but rather as a branched rough path.

Rough path theory allows to give a pathwise notion of solution to (stochastic) differential equations of the form

:$\backslash mathrm\{d\}Y\_t\; =\; b(Y\_t)\backslash ,\; \backslash mathrm\{d\}t\; +\; \backslash sigma(Y\_t)\; \backslash ,\; \backslash mathrm\{d\}X\_t$

provided that we can construct a rough path which is almost surely a rough path lift of the multidimensional stochastic process $X\_t$ and that the drift $b$ and the volatility $\backslash sigma$ are sufficiently smooth (see the section on the Universal Limit Theorem).

There are many examples of Markov processes, Gaussian processes, and other processes that can be enhanced as rough paths.{{cite book|last1=Friz|first1=Peter K.|last2=Victoir|first2=Nicolas|title=Multidimensional Stochastic Processes as Rough Paths: Theory and Applications|date=2010|publisher=Cambridge University Press|edition=Cambridge Studies in Advanced Mathematics}}

There are, in particular, many results on the solution to differential equation driven by fractional Brownian motion that have been proved using a combination of Malliavin calculus and rough path theory. In fact, it has been proved recently that the solution to controlled differential equation driven by a class of Gaussian processes, which includes fractional Brownian motion with Hurst parameter $H>\backslash frac\{1\}\{4\}$, has a smooth density under the Hörmander's condition on the vector fields.{{cite journal|last1=Cass|first1=Thomas|last2=Friz|first2=Peter|title=Densities for rough differential equations under Hörmander's condition|journal=Annals of Mathematics|date=2010|doi=10.4007/annals.2010.171.2115 |volume=171|issue=3|pages=2115–2141|arxiv=0708.3730|s2cid=17276607}}

{{cite journal|last1=Cass|first1=Thomas|last2=Hairer|first2=Martin|last3=Litterer|first3=Christian|last4=Tindel|first4=Samy|title=Smoothness of the density for solutions to Gaussian rough differential equations|journal=The Annals of Probability|date=2015|doi=10.1214/13-AOP896|volume=43|pages=188–239|arxiv=1209.3100|s2cid=17308794}}

Let $L(V,W)$ denote the space of bounded linear maps from a Banach space $V$ to another Banach space $W$.

Let $B\_t$ be a $d$-dimensional standard Brownian motion. Let $b:\backslash mathbb\{R\}^n\backslash rightarrow\; \backslash mathbb\{R\}^d$ and $\backslash sigma:\backslash mathbb\{R\}^n\backslash rightarrow\; L(\backslash mathbb\{R\}^d,\backslash mathbb\{R\}^n)$ be twice-differentiable functions and whose second derivatives are $\backslash alpha$-Hölder for some $\backslash alpha>0$.

Let $X^\{\backslash varepsilon\}$ be the unique solution to the stochastic differential equation

: $\backslash mathrm\{d\}X^\{\backslash varepsilon\}\; =\; b(X^\{\backslash epsilon\}\_t)\; \backslash ,\; \backslash mathrm\{d\}t\; +\; \backslash sqrt\{\backslash varepsilon\}\; \backslash sigma(X^\backslash varepsilon)\; \backslash circ\; \backslash mathrm\{d\}B\_t;\backslash ,X^\{\backslash varepsilon\}=a,$

where $\backslash circ$ denotes Stratonovich integration.

The Freidlin Wentzell's large deviation theory aims to study the asymptotic behavior, as $\backslash epsilon\; \backslash rightarrow\; 0$, of $\backslash mathbb\{P\}[X^\backslash varepsilon\; \backslash in\; F]$ for closed or open sets $F$ with respect to the uniform topology.

The Universal Limit Theorem guarantees that the Itô map sending the control path $(t,\backslash sqrt\{\backslash varepsilon\}B\_t)$ to the solution $X^\backslash varepsilon$ is a continuous map from the $p$-variation topology to the $p$-variation topology (and hence the uniform topology). Therefore, the Contraction principle in large deviations theory reduces Freidlin–Wentzell's problem to demonstrating the large deviation principle for $(t,\backslash sqrt\{\backslash varepsilon\}B\_t)$ in the $p$-variation topology.

This strategy can be applied to not just differential equations driven by the Brownian motion but also to the differential equations driven any stochastic processes which can be enhanced as rough paths, such as fractional Brownian motion.

Once again, let $B\_t$ be a $d$-dimensional Brownian motion. Assume that the drift term $b$ and the volatility term $\backslash sigma$ has sufficient regularity so that the stochastic differential equation

:$\backslash mathrm\{d\}\backslash phi\_\{s,t\}(x)\; =\; b(\backslash phi\_\{s,t\}(x))\; \backslash ,\; \backslash mathrm\{d\}t\; +\; \backslash sigma\{(\backslash phi\_\{s,t\}(x))\}\; \backslash ,\; \backslash mathrm\{d\}B\_t;\; X\_s=x$

has a unique solution in the sense of rough path. A basic question in the theory of stochastic flow is whether the flow map $\backslash phi\_\{s,t\}(x)$ exists and satisfy the cocyclic property that for all $s\backslash leq\; u\backslash leq\; t$,

:$\backslash phi\_\{u,t\}(\backslash phi\_\{s,u\}(x))=\backslash phi\_\{s,t\}(x)$

outside a null set independent of $s,u,t$.

The Universal Limit Theorem once again reduces this problem to whether the Brownian rough path $\backslash mathbf\{B\_\{s,t\}\}$ exists and satisfies the multiplicative property that for all $s\backslash leq\; u\; \backslash leq\; t$,

:$\backslash mathbf\{B\}\_\{s,u\}\; \backslash otimes\; \backslash mathbf\{B\}\_\{u,t\}\; =\; \backslash mathbf\{B\}\_\{s,t\}$

outside a null set independent of $s$, $u$ and $t$.

In fact, rough path theory gives the existence and uniqueness of $\backslash phi\_\{s,t\}(x)$ not only outside a null set independent of $s$,$t$ and $x$ but also of the drift $b$ and the volatility $\backslash sigma$.

As in the case of Freidlin–Wentzell theory, this strategy holds not just for differential equations driven by the Brownian motion but to any stochastic processes that can be enhanced as rough paths.

Controlled rough paths, introduced by M. Gubinelli,{{cite Q|Q56689330|author1=Gubinelli, Massimiliano}} are paths $\backslash mathbf\{Y\}$ for which the rough integral

: $\backslash int\_s^t\; \backslash mathbf\{Y\}\_u\; \backslash ,\; \backslash mathrm\{d\}X\_u$

can be defined for a given geometric rough path $X$.

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

Given a $p$-geometric rough path

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

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

: $\backslash Vert\; \backslash mathbf\{Y\}^\{j\}\_s\; \backslash Vert\backslash leq\; M$

and

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

Let $\backslash mathbf\{X\}=(1,\backslash mathbf\{X\}^\{1\},\backslash ldots,\backslash mathbf\{X\}^\{\backslash lfloor\; p\; \backslash rfloor\})$ be a $p$-geometric rough path satisfying the Hölder condition that there exists $M>0$, for all $0\backslash leq\; s\backslash leq\; t\; \backslash leq\; 1$ and all $j=1,,2,\backslash ldots,\backslash lfloor\; p\; \backslash rfloor$,

:$\backslash Vert\; \backslash mathbf\{X\}^j\_\{s,t\}\; \backslash Vert\; \backslash leq\; M(t-s)^\{\backslash frac\{j\}\{p\}\},$

where $\backslash mathbf\{X\}^j$ denotes the $j$-th tensor component of $\backslash mathbf\{X\}$.

Let $\backslash gamma\backslash geq\; 1$. Let $f:\backslash mathbb\{R\}^\{d\}\backslash rightarrow\; \backslash mathbb\{R\}^\{n\}$ be an $\backslash lfloor\; \backslash gamma\; \backslash rfloor$-times differentiable function and the $\backslash lfloor\; \backslash gamma\; \backslash rfloor$-th derivative is $\backslash gamma\; -\; \backslash lfloor\; \backslash gamma\; \backslash rfloor$ Hölder, then

:$(f(\backslash mathbf\{X\}^1\_s),Df(\backslash mathbf\{X\}^1\_s),\backslash ldots,D^\{\backslash lfloor\; \backslash gamma\; \backslash rfloor\}\; f(\backslash mathbf\{X\}^1\_s))$

is a $\backslash gamma$-controlled path.

If $\backslash mathbf\{Y\}$ is a $\backslash gamma$-controlled path where $\backslash gamma>p-1$, then

: $\backslash int\_s^t\; \backslash mathbf\{Y\}\_u\; \backslash ,\; \backslash mathrm\{d\}X\_u$

is defined and the path

: $\backslash left(\; \backslash int\_s^t\; \backslash mathbf\{Y\}\_u\; \backslash ,\; \backslash mathrm\{d\}X\_u,\; \backslash mathbf\{Y\}^0\_s,\; \backslash mathbf\{Y\}^1\_s,\; \backslash ldots,\; \backslash mathbf\{Y\}^\{\backslash lfloor\; \backslash gamma-1\; \backslash rfloor\}\_s\; \backslash right)$

is a $\backslash gamma$-controlled path.

Let $V:\backslash mathbb\{R\}^n\; \backslash rightarrow\; L(\backslash mathbb\{R\}^d,\backslash mathbb\{R\}^n)$ be functions that has at least $\backslash lfloor\; \backslash gamma\; \backslash rfloor$ derivatives and the $\backslash lfloor\; \backslash gamma\; \backslash rfloor$-th derivatives are $\backslash gamma-\backslash lfloor\; \backslash gamma\; \backslash rfloor$-Hölder continuous for some $\backslash gamma\; >\; p$. Let $Y$ be the solution to the differential equation

:$\backslash mathrm\{d\}\; Y\_t\; =\; V(Y\_t)\; \backslash ,\; \backslash mathrm\{d\}X\_t\; .$

Define

:$\backslash frac\{\backslash mathrm\{d\}Y\}\{\backslash mathrm\{d\}X\}(\backslash cdot)=V(\backslash cdot);$

:$\backslash frac\{\backslash mathrm\{d\}^\{r+1\}Y\}\; \{\backslash mathrm\{d\}^\{r+1\}X\}(\backslash cdot)\; =\; D\; \backslash left(\; \backslash frac\{\backslash mathrm\{d\}^r\; Y\}\{\backslash mathrm\{d\}^r\; X\}\; \backslash right)\; (\backslash cdot)\; V(\backslash cdot),$

where $D$ denotes the derivative operator, then

: $\backslash left(Y\_t,\; \backslash frac\{\backslash mathrm\{d\}Y\}\{\backslash mathrm\{d\}X\}(Y\_t),\; \backslash frac\{\backslash mathrm\{d\}^2Y\}\{\backslash mathrm\{d\}^2X\}(Y\_t),\; \backslash ldots,\; \backslash frac\{\backslash mathrm\{d\}^\{\backslash lfloor\; \backslash gamma\; \backslash rfloor\}Y\}\{\backslash mathrm\{d\}^\{\backslash lfloor\; \backslash gamma\; \backslash rfloor\}X\}(Y\_t)\backslash right)$

is a $\backslash gamma$-controlled path.

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

:

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

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

:

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

:

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

The signature satisfies Chen's identity,{{cite journal |last1=Chen |first1=Kuo-Tsai |date=1954 |title=Iterated Integrals and Exponential Homomorphisms |journal=Proceedings of the London Mathematical Society |volume=s3-4 |pages=502–512 |doi=10.1112/plms/s3-4.1.502}} that

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

for all $s\; \backslash leq\; u\; \backslash leq\; t$.

The set of paths whose signature is the trivial sequence, or more precisely,

: $S(\backslash mathbf\{X\})\_\{0,1\}\; =\; (1,0,0,\backslash ldots)$

can be completely characterized using the idea of tree-like path.

A $p$-geometric rough path is tree-like if there exists a continuous function $h:[0,1]\backslash rightarrow\; [0,\backslash infty)$ such that $h(0)=h(1)=0$ and for all $j=1,\backslash ldots,\backslash lfloor\; p\; \backslash rfloor$ and all $0\backslash leq\; s\; \backslash leq\; t\backslash leq\; 1$,

: $\backslash Vert\; \backslash mathbf\{X\}^j\_\{s,t\}\; \backslash Vert^p\; \backslash leq\; h(t)+h(s)-2\backslash inf\_\{u\backslash in\; [s,t]\}h(u)$

where $\backslash mathbf\{X\}^\{j\}$ denotes the $j$-th tensor component of $\backslash mathbf\{X\}$.

A geometric rough path $\backslash mathbf\{X\}$ satisfies $S(\backslash mathbf\{X\})\_\{0,1\}=(1,0,\backslash ldots)$ if and only if $\backslash mathbf\{X\}$ is tree-like.{{cite journal|last1=Hambly|first1=Ben|last2=Lyons|first2=Terry|title=Uniqueness for the signature of a path of bounded variation and the reduced path group|journal=Annals of Mathematics|date=2010|doi=10.4007/annals.2010.171.109|volume=171|pages=109–167|arxiv=math/0507536|s2cid=15915599}}{{cite journal|last1=Boedihardjo|first1=Horatio|last2=Geng|first2=Xi|last3=Lyons|first3=Terry|last4=Yang|first4=Danyu|title=The signature of a rough path: Uniqueness|journal=Advances in Mathematics|date=2016|doi=10.1016/j.aim.2016.02.011|doi-access=free|volume=293|pages=720–737|arxiv=1406.7871|s2cid=3634324}}

Given the signature of a path, it is possible to reconstruct the unique path that has no tree-like pieces.{{cite journal

| last1=Lyons | first1=Terry

| last2=Xu | first2=Weijun

| title=Inverting the signature of a path

| journal=Journal of the European Mathematical Society

| year=2018

| volume=20

| issue=7

| pages=1655–1687

| doi=10.4171/JEMS/796

| arxiv=1406.7833

| s2cid=67847036

}}{{cite journal|last1=Geng|first1=Xi|title=Reconstruction for the Signature of a Rough Path|journal=Proceedings of the London Mathematical Society|arxiv=1508.06890|date=2016|volume=114|issue=3|pages=495–526|doi=10.1112/plms.12013|s2cid=3641736}}

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.{{cite journal|last1=Cass|first1=Thomas|last2=Driver|first2=Bruce|last3=Lim|first3=Nengli|last4=Litterer|first4=Christian|title=On the integration of weakly geometric rough paths|journal=Journal of the Mathematical Society of Japan}}

{{Reflist}}

• {{cite book |first=Antoine |last=Lejay |year=2009 |chapter=Yet Another Introduction to Rough Paths |pages=1–101 |editor1-first=Catherine |editor1-last=Donati-Martin |editor2-first=Michel |editor2-last=Émery |editor3-first=Alain |editor3-last=Rouault |editor4-first=Christophe |editor4-last=Stricker |display-editors=1 |title=Séminaire de Probabilités XLII |series=Lecture Notes in Mathematics |volume=1979 |publisher=Springer |location=Berlin |isbn=978-3-642-01762-9 }}

Category:Differential equations

Category:Stochastic processes