Motivation
Definition of a rough path
Universal limit theorem
Examples of rough paths
Brownian motion
. This geometric rough path is called the Stratonovich Brownian rough path.
Fractional Brownian motion
. This limiting geometric rough path can be used to make sense of differential equations driven by fractional Brownian motion with Hurst parameter H>\frac. When 0, it turns out that the above limit along dyadic approximations does not converge in 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. Non-uniqueness of enhancement In general, let (X_t)_ be a \mathbb^d-valued stochastic process. If one can construct, almost surely, functions (s,t)\rightarrow \mathbf^_ \in \big(\mathbb^d\big)^ so that : \mathbf:(s,t)\rightarrow (1,X_t-X_s,\mathbf^2_,\ldots,\mathbf^_) is a p-geometric rough path, then \mathbf_ 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 :\mathrm Y^i_t = \sum^d_ V^i_j(Y_t) \, \mathrm 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. 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. This implies that the Stratonovich calculus is not the only theory of stochastic calculus that satisfies the classical product rule : \mathrm(X_t\cdot Y_t) = X_t \, \mathrm Y_t+Y_t \, \mathrm 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 Itô calculus, named after Kiyosi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance and stochastic differential equations. The centra ... does not come directly from enhancing Brownian motion as a geometric rough path, but rather as a branched rough path. Applications in stochastic analysis Stochastic differential equations driven by non-semimartingales Rough path theory allows to give a pathwise notion of solution to (stochastic) differential equations of the form : \mathrmY_t = b(Y_t)\, \mathrmt + \sigma(Y_t) \, \mathrmX_t provided that the multidimensional stochastic process X_t can be almost surely enhanced as a rough path and that the drift b and the volatility \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. 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 In probability theory and related fields, Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to stochastic processes. In particular, it allows ... 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>\frac, has a smooth density under the Hörmander's condition In mathematics, Hörmander's condition is a property of vector fields that, if satisfied, has many useful consequences in the theory of partial and stochastic differential equations. The condition is named after the Swedish mathematician Lars Hör ... on the vector fields. Freidlin–Wentzell's large deviation theory 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:\mathbb^n\rightarrow \mathbb^d and \sigma:\mathbb^n\rightarrow L(\mathbb^d,\mathbb^n) be twice-differentiable functions and whose second derivatives are \alpha-Hölder for some \alpha>0. Let X^ be the unique solution to the stochastic differential equation : \mathrmX^ = b(X^_t) \, \mathrmt + \sqrt \sigma(X^\varepsilon) \circ \mathrmB_t;\,X^=a, where \circ denotes Stratonovich integration. The Freidlin Wentzell's large deviation theory aims to study the asymptotic behavior, as \epsilon \rightarrow 0, of \mathbb ^\varepsilon \in F/math> 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,\sqrtB_t) to the solution X^\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,\sqrtB_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. Stochastic flow Once again, let B_t be a d-dimensional Brownian motion. Assume that the drift term b and the volatility term \sigma has sufficient regularity so that the stochastic differential equation :\mathrm\phi_(x) = b(\phi_(x)) \, \mathrmt + \sigma \, \mathrmB_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 \phi_(x) exists and satisfy the cocyclic property that for all s\leq u\leq t, : \phi_(\phi_(x))=\phi_(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 \mathbf exists and satisfies the multiplicative property that for all s\leq u \leq t, : \mathbf_ \otimes \mathbf_ = \mathbf_ outside a null set independent of s, u and t. In fact, rough path theory gives the existence and uniqueness of \phi_(x) not only outside a null set independent of s,t and x but also of the drift b and the volatility \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 path Controlled rough paths, introduced by M. Gubinelli, are paths \mathbf for which the rough integral : \int_s^t \mathbf_u \, \mathrmX_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 : \mathbf = (1,\mathbf^1, \ldots, \mathbf^) on \mathbb^, a \gamma-controlled path is a function \mathbf_s =(\mathbf^0_s,\mathbf^1_s, \ldots, \mathbf^_) such that \mathbf^j: ,1\rightarrow L((\mathbb^d)^, \mathbb^n) and that there exists M>0 such that for all 0\leq s\leq t\leq 1 and j=0,1,\ldots,\lfloor \gamma \rfloor, : \Vert \mathbf^_s \Vert\leq M and : \left\, \mathbf^j_t - \sum_^ \mathbf_s^ \mathbf^i_ \right\, \leq M, t-s, ^. Example: Lip(''γ'') function Let \mathbf=(1,\mathbf^,\ldots,\mathbf^) be a p-geometric rough path satisfying the Hölder condition In mathematics, a real or complex-valued function ''f'' on ''d''-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants ''C'', α > 0, such that : , f(x) - f(y) , \leq C\, ... that there exists M>0, for all 0\leq s\leq t \leq 1 and all j=1,,2,\ldots,\lfloor p \rfloor, : \Vert \mathbf^j_ \Vert \leq M(t-s)^, where \mathbf^j denotes the j-th tensor component of \mathbf. Let \gamma\geq 1 . Let f:\mathbb^\rightarrow \mathbb^ be an \lfloor \gamma \rfloor-times differentiable function and the \lfloor \gamma \rfloor-th derivative is \gamma - \lfloor \gamma \rfloor Hölder, then : (f(\mathbf^1_s),Df(\mathbf^1_s),\ldots,D^ f(\mathbf^1_s)) is a \gamma-controlled path. Integral of a controlled path is a controlled path If \mathbf is a \gamma-controlled path where \gamma>p-1, then : \int_s^t \mathbf_u \, \mathrmX_u is defined and the path : \left( \int_s^t \mathbf_u \, \mathrmX_u, \mathbf^0_s, \mathbf^1_s, \ldots, \mathbf^_s \right) is a \gamma-controlled path. Solution to controlled differential equation is a controlled path Let V:\mathbb^n \rightarrow L(\mathbb^d,\mathbb^n) be functions that has at least \lfloor \gamma \rfloor derivatives and the \lfloor \gamma \rfloor-th derivatives are \gamma-\lfloor \gamma \rfloor-Hölder continuous for some \gamma > p . Let Y be the solution to the differential equation :\mathrm Y_t = V(Y_t) \, \mathrmX_t . Define : \frac(\cdot)=V(\cdot); : \frac (\cdot) = D \left( \frac \right) (\cdot) V(\cdot), where D denotes the derivative operator, then : \left(Y_t, \frac(Y_t), \frac(Y_t), \ldots, \frac(Y_t)\right) is a \gamma-controlled path. Signature Let X: ,1rightarrow \mathbb^ be a continuous function with finite total variation. Define : S(X)_= \left( 1,\int_ \mathrmX_,\int_ \mathrmX_ \otimes \mathrmX_, \ldots, \int_ \mathrmX_ \otimes \cdots \otimes\mathrm X_,\ldots\right). The signature of a path is defined to be S(X)_. The signature can also be defined for geometric rough paths. Let \mathbf be a geometric rough path and let \mathbf(n) be a sequence of paths with finite total variation such that : \mathbf(n)_= \left(1, \int_ \, \mathrmX(n)_, \ldots, \int_ \, \mathrm X(n)_ \otimes \cdots \otimes \mathrm X(n)_\right). converges in the p-variation metric to \mathbf. Then : \int_ \, \mathrmX(n)_\otimes \cdots \otimes \mathrmX(n)_ converges as n\rightarrow \infty for each N. The signature of the geometric rough path \mathbf can be defined as the limit of S(X(n))_ as n\rightarrow \infty. The signature satisfies Chen's identity, that : S(\mathbf)_\otimes S(\mathbf)_=S(\mathbf)_ for all s \leq u \leq t. Kernel of the signature transform The set of paths whose signature is the trivial sequence, or more precisely, : S(\mathbf)_ = (1,0,0,\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: ,1rightarrow [0,\infty) such that h(0)=h(1)=0 and for all j=1,\ldots,\lfloor p \rfloor and all 0\leq s \leq t\leq 1, : \Vert \mathbf^j_ \Vert^p \leq h(t)+h(s)-2\inf_h(u) where \mathbf^ denotes the j-th tensor component of \mathbf. A geometric rough path \mathbf satisfies S(\mathbf)_=(1,0,\ldots) if and only if \mathbf is tree-like. Given the signature of a path, it is possible to reconstruct the unique path that has no tree-like pieces. 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. References {{Reflist Differential equations Stochastic processes
Non-uniqueness of enhancement
Applications in stochastic analysis
Stochastic differential equations driven by non-semimartingales
Freidlin–Wentzell's large deviation theory
Stochastic flow
Controlled rough path
Example: Lip(''γ'') function
Integral of a controlled path is a controlled path
Solution to controlled differential equation is a controlled path
Signature
Kernel of the signature transform
Infinite dimensions
References