Rough path

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

${\displaystyle \mathbf {B} _{s,t}=\left(1,\int _{s<s_{1}<t}\circ \mathrm {d} B_{s_{1}},\int _{s<s_{1}<s_{2}<t}\circ \mathrm {d} B_{s_{1}}\otimes \circ \mathrm {d} B_{s_{2}}\right)}$

is a ${\displaystyle p}$-geometric rough path for any ${\displaystyle 2<p<3}$. This geometric rough path is called the Stratonovich Brownian rough path.

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

${\displaystyle {\begin{aligned}\mathbf {B} _{H}^{m}(s,t)=\left(1,\int _{s<s_{1}<t}\right.&\mathrm {d} B_{H}^{m}(s_{1}),\int _{s<s_{1}<s_{2}<t}\,\mathrm {d} B_{H}^{m}(s_{1})\otimes \mathrm {d} B_{H}^{m}(s_{2}),\\&\left.\int _{s<s_{1}<s_{2}<s_{3}<t}\mathrm {d} B_{H}^{m}(s_{1})\otimes \mathrm {d} B_{H}^{m}(s_{2})\otimes \mathrm {d} B_{H}^{m}(s_{3})\right)\end{aligned}}}$

converges almost surely in the ${\displaystyle p}$-variation metric to a ${\displaystyle p}$-geometric rough path for ${\displaystyle {\frac {1}{H}}<p}$.[13] This limiting geometric rough path can be used to make sense of differential equations driven by fractional Brownian motion with Hurst parameter ${\displaystyle H>{\frac {1}{4}}}$. When ${\displaystyle 0<H\leq {\frac {1}{4}}}$, it turns out that the above limit along dyadic approximations does not converge in ${\displaystyle p}$-variation. However, one can of course still make sense of differential equations provided one exhibits a rough path lift, existence of such a (non-unique) lift is a consequence of the Lyons–Victoir extension theorem.

In general, let ${\displaystyle (X_{t})_{t\geq 0}}$ be a ${\displaystyle \mathbb {R} ^{d}}$-valued stochastic process. If one can construct, almost surely, functions ${\displaystyle (s,t)\rightarrow \mathbf {X} _{s,t}^{j}\in {\big (}\mathbb {R} ^{d}{\big )}^{\otimes j}}$ so that

${\displaystyle \mathbf {X}$ :(s,t)\rightarrow (1,X_{t}-X_{s},\mathbf {X} _{s,t}^{2},\ldots ,\mathbf {X} _{s,t}^{\lfloor p\rfloor })}

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

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

for sufficiently regular vector fields ${\displaystyle V_{j}^{i}.}$

Note that every stochastic process (even if it is a deterministic path) can have more than one (in fact, uncountably many) possible enhancements.[14] 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.[15] This implies that the Stratonovich calculus is not the only theory of stochastic calculus that satisfies the classical product rule

${\displaystyle \mathrm {d} (X_{t}\cdot Y_{t})=X_{t}\,\mathrm {d} Y_{t}+Y_{t}\,\mathrm {d} X_{t}.}$

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

${\displaystyle \mathrm {d} Y_{t}=b(Y_{t})\,\mathrm {d} t+\sigma (Y_{t})\,\mathrm {d} X_{t}}$

provided that the multidimensional stochastic process ${\displaystyle X_{t}}$ can be almost surely enhanced as a rough path and that the drift ${\displaystyle b}$ and the volatility ${\displaystyle \sigma }$ 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.[16]

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 ${\displaystyle H>{\frac {1}{4}}}$, has a smooth density under the Hörmander's condition on the vector fields.[17] [18]

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

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

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

${\displaystyle \mathrm {d} X^{\varepsilon }=b(X_{t}^{\epsilon })\,\mathrm {d} t+{\sqrt {\varepsilon }}\sigma (X^{\varepsilon })\circ \mathrm {d} B_{t};\,X^{\varepsilon }=a,}$

where ${\displaystyle \circ }$ denotes Stratonovich integration.

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

The Universal Limit Theorem guarantees that the Itô map sending the control path ${\displaystyle (t,{\sqrt {\varepsilon }}B_{t})}$ to the solution ${\displaystyle X^{\varepsilon }}$ is a continuous map from the ${\displaystyle p}$-variation topology to the ${\displaystyle p}$-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 ${\displaystyle (t,{\sqrt {\varepsilon }}B_{t})}$ in the ${\displaystyle p}$-variation topology.[10]

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 ${\displaystyle B_{t}}$ be a ${\displaystyle d}$-dimensional Brownian motion. Assume that the drift term ${\displaystyle b}$ and the volatility term ${\displaystyle \sigma }$ has sufficient regularity so that the stochastic differential equation

${\displaystyle \mathrm {d} \phi _{s,t}(x)=b(\phi _{s,t}(x))\,\mathrm {d} t+\sigma {(\phi _{s,t}(x))}\,\mathrm {d} B_{t};X_{s}=x}$

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

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

outside a null set independent of ${\displaystyle s,u,t}$.

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

${\displaystyle \mathbf {B} _{s,u}\otimes \mathbf {B} _{u,t}=\mathbf {B} _{s,t}}$

outside a null set independent of ${\displaystyle s}$, ${\displaystyle u}$ and ${\displaystyle t}$.

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

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.

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

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

where ${\displaystyle \mathbf {X} ^{j}}$ denotes the ${\displaystyle j}$-th tensor component of ${\displaystyle \mathbf {X} }$. Let ${\displaystyle \gamma \geq 1}$. Let ${\displaystyle f:\mathbb {R} ^{d}\rightarrow \mathbb {R} ^{n}}$ be an ${\displaystyle \lfloor \gamma \rfloor }$-times differentiable function and the ${\displaystyle \lfloor \gamma \rfloor }$-th derivative is ${\displaystyle \gamma -\lfloor \gamma \rfloor }$ Hölder, then

${\displaystyle (f(\mathbf {X} _{s}^{1}),Df(\mathbf {X} _{s}^{1}),\ldots ,D^{\lfloor \gamma \rfloor }f(\mathbf {X} _{s}^{1}))}$

is a ${\displaystyle \gamma }$-controlled path.

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

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

is defined and the path

${\displaystyle \left(\int _{s}^{t}\mathbf {Y} _{u}\,\mathrm {d} X_{u},\mathbf {Y} _{s}^{0},\mathbf {Y} _{s}^{1},\ldots ,\mathbf {Y} _{s}^{\lfloor \gamma -1\rfloor }\right)}$

is a ${\displaystyle \gamma }$-controlled path.

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

${\displaystyle \mathrm {d} Y_{t}=V(Y_{t})\,\mathrm {d} X_{t}.}$

Define

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

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

where ${\displaystyle D}$ denotes the derivative operator, then

${\displaystyle \left(Y_{t},{\frac {\mathrm {d} Y}{\mathrm {d} X}}(Y_{t}),{\frac {\mathrm {d} ^{2}Y}{\mathrm {d} ^{2}X}}(Y_{t}),\ldots ,{\frac {\mathrm {d} ^{\lfloor \gamma \rfloor }Y}{\mathrm {d} ^{\lfloor \gamma \rfloor }X}}(Y_{t})\right)}$

is a ${\displaystyle \gamma }$-controlled path.

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

${\displaystyle S(\mathbf {X} )_{0,1}=(1,0,0,\ldots )}$

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

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

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

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

A geometric rough path ${\displaystyle \mathbf {X} }$ satisfies ${\displaystyle S(\mathbf {X} )_{0,1}=(1,0,\ldots )}$ if and only if ${\displaystyle \mathbf {X} }$ is tree-like.[20][21]

Given the signature of a path, it is possible to reconstruct the unique path that has no tree-like pieces.[22][23]