Rough Path Theory Research Articles

Smooth approximation of stochastic differential equations

Consider an It\^{o} process $X$ satisfying the stochastic differential equation $dX=a(X)\,dt+b(X)\,dW$ where $a,b$ are smooth and $W$ is a multidimensional Brownian motion. Suppose that $W_n$ has smooth sample paths and that $W_n$ converges weakly to $W$. A central question in stochastic analysis is to understand the limiting behavior of solutions $X_n$ to the ordinary differential equation $dX_n=a(X_n)\,dt+b(X_n)\,dW_n$. The classical Wong--Zakai theorem gives sufficient conditions under which $X_n$ converges weakly to $X$ provided that the stochastic integral $\int b(X)\,dW$ is given the Stratonovich interpretation. The sufficient conditions are automatic in one dimension, but in higher dimensions the correct interpretation of $\int b(X)\,dW$ depends sensitively on how the smooth approximation $W_n$ is chosen. In applications, a natural class of smooth approximations arise by setting $W_n(t)=n^{-1/2}\int_0^{nt}v\circ\phi_s\,ds$ where $\phi_t$ is a flow (generated, e.g., by an ordinary differential equation) and $v$ is a mean zero observable. Under mild conditions on $\phi_t$, we give a definitive answer to the interpretation question for the stochastic integral $\int b(X)\,dW$. Our theory applies to Anosov or Axiom A flows $\phi_t$, as well as to a large class of nonuniformly hyperbolic flows (including the one defined by the well-known Lorenz equations) and our main results do not require any mixing assumptions on $\phi_t$. The methods used in this paper are a combination of rough path theory and smooth ergodic theory.

On the pathwise approximation of stochastic differential equations

We consider one-step methods for integrating stochastic differential equations and prove pathwise convergence using ideas from rough path theory. In contrast to alternative theories of pathwise convergence, no knowledge is required of convergence in pth mean and the analysis starts from a pathwise bound on the sum of the truncation errors. We show how the theory is applied to the Euler–Maruyama method with fixed and adaptive time-stepping strategies. The assumption on the truncation errors suggests an error-control strategy and we implement this as an adaptive time-stepping Euler–Maruyama method using bounded diffusions. We prove the adaptive method converges and show some computational experiments.

Pathwise Itô calculus for rough paths and rough PDEs with path dependent coefficients

This paper introduces path derivatives, in the spirit of Dupire’s functional Itô calculus, for controlled rough paths in rough path theory with possibly non-geometric rough paths. We next study rough PDEs with coefficients depending on the rough path itself, which corresponds to stochastic PDEs with random coefficients. Such coefficients are less regular in the time variable, which is not covered in the existing literature. The results are useful for studying viscosity solutions of stochastic PDEs.

The theory of rough paths via one-forms and the extension of an argument of Schwartz to rough differential equations

We give an overview of the recent approach to the integration of rough paths that reduces the problem to classical Young integration. As an application, we extend an argument of Schwartz to rough differential equations, and prove the existence, uniqueness and continuity of the solution, which is applicable when the driving path takes values in nilpotent Lie group or Butcher group.

Expected signature of Brownian motion up to the first exit time from a bounded domain

The signature of a path provides a top down description of the path in terms of its effects as a control [Differential Equations Driven by Rough Paths (2007) Springer]. The signature transforms a path into a group-like element in the tensor algebra and is an essential object in rough path theory. The expected signature of a stochastic process plays a similar role to that played by the characteristic function of a random variable. In [Chevyrev (2013)], it is proved that under certain boundedness conditions, the expected value of a random signature already determines the law of this random signature. It becomes of great interest to be able to compute examples of expected signatures and obtain the upper bounds for the decay rates of expected signatures. For instance, the computation for Brownian motion on $[0,1]$ leads to the “cubature on Wiener space” methodology [Lyons and Victoir, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004) 169–198]. In this paper we fix a bounded domain $\Gamma$ in a Euclidean space $E$ and study the expected signature of a Brownian path starting at $z\in\Gamma$ and stopped at the first exit time from $\Gamma$. We denote this tensor series valued function by $\Phi_{\Gamma}(z)$ and focus on the case $E=\mathbb{R}^{d}$. We show that $\Phi_{\Gamma}(z)$ satisfies an elliptic PDE system and a boundary condition. The equations determining $\Phi_{\Gamma}$ can be recursively solved; by an iterative application of Sobolev estimates we are able, under certain smoothness and boundedness condition of the domain $\Gamma$, to prove geometric bounds for the terms in $\Phi_{\Gamma}(z)$. However, there is still a gap and we have not shown that $\Phi_{\Gamma}(z)$ determines the law of the signature of this stopped Brownian motion even if $\Gamma$ is a unit ball.

Rough paths and 1d SDE with a time dependent distributional drift: application to polymers

Motivated by the recent advances in the theory of stochastic partial differential equations involving nonlinear functions of distributions, like the Kardar-Parisi-Zhang (KPZ) equation, we reconsider the unique solvability of one-dimensional stochastic differential equations, the drift of which is a distribution, by means of rough paths theory. Existence and uniqueness are established in the weak sense when the drift reads as the derivative of a Holder continuous function. Regularity of the drift part is investigated carefully and a related stochastic calculus is also proposed, which makes the structure of the solutions more explicit than within the earlier framework of Dirichlet processes.

Large deviation principle of Freidlin-Wentzell type for pinned diffusion processes

Since T. Lyons invented rough path theory, one of its most successful applications is a new proof of Freidlin-Wentzell’s large deviation principle for diffusion processes. In this paper we extend this method to the case of pinned diffusion processes under a mild ellipticity assumption. Besides rough path theory, our main tool is quasi-sure analysis, which is a kind of potential theory in Malliavin calculus.

Stochastic Analysis: Around the KPZ Universality Class

The Gaussian distribution is the “universal” distribution arising in a huge variety of contexts that describes the compound effect of the random fluctuations of many independent (or weakly dependent) sources of randomness that are combined in a (close to) additive way. While this has been very well understood for a long time, the last few years have seen an explosion of results around the “KPZ universality class”, which contains many systems where strongly interacting individual components are combined in a highly non-linear way. In this class, which is still rather poorly understood from a mathematical perspective, fluctuations typically exhibit scaling exponent 1/3 instead of the exponent 1/2 familiar from the central limit theorem and limiting distributions are of Tracy-Widom type rather than Gaussian. This workshop brought together outstanding researchers from a variety of mathematical backgrounds whose areas of research are linked to the understanding of the KPZ equation and universality class. While there are strong links between their motivations, the techniques used by these researchers span a large swath of mathematics, ranging from purely algebraic techniques to renormalisation theory, stochastic analysis, random matrix theory, classical probability theory, orthogonal polynomials, the theory of rough paths, etc.

Rough path theory and stochastic analysis

Rough path theory and stochastic analysis

The stochastic filtering problem: a brief historical account

Onwards from the mid-twentieth century, the stochastic filtering problem has caught the attention of thousands of mathematicians, engineers, statisticians, and computer scientists. Its applications span the whole spectrum of human endeavour, including satellite tracking, credit risk estimation, human genome analysis, and speech recognition. Stochastic filtering has engendered a surprising number of mathematical techniques for its treatment and has played an important role in the development of new research areas, including stochastic partial differential equations, stochastic geometry, rough paths theory, and Malliavin calculus. It also spearheaded research in areas of classical mathematics, such as Lie algebras, control theory, and information theory. The aim of this paper is to give a brief historical account of the subject concentrating on the continuous-time framework.

The stochastic filtering problem: a brief historical account

Onwards from the mid-twentieth century, the stochastic filtering problem has caught the attention of thousands of mathematicians, engineers, statisticians, and computer scientists. Its applications span the whole spectrum of human endeavour, including satellite tracking, credit risk estimation, human genome analysis, and speech recognition. Stochastic filtering has engendered a surprising number of mathematical techniques for its treatment and has played an important role in the development of new research areas, including stochastic partial differential equations, stochastic geometry, rough paths theory, and Malliavin calculus. It also spearheaded research in areas of classical mathematics, such as Lie algebras, control theory, and information theory. The aim of this paper is to give a brief historical account of the subject concentrating on the continuous-time framework.

Robust filtering: Correlated noise and multidimensional observation

In the late seventies, Clark [In Communication Systems and Random Process Theory (Proc. 2nd NATO Advanced Study Inst., Darlington, 1977) (1978) 721–734, Sijthoff & Noordhoff] pointed out that it would be natural for $\pi_{t}$, the solution of the stochastic filtering problem, to depend continuously on the observed data $Y=\{Y_{s},s\in[0,t]\}$. Indeed, if the signal and the observation noise are independent one can show that, for any suitably chosen test function $f$, there exists a continuous map $\theta^{f}_{t}$, defined on the space of continuous paths $C([0,t],\mathbb{R} ^{d})$ endowed with the uniform convergence topology such that $\pi_{t}(f)=\theta^{f}_{t}(Y)$, almost surely; see, for example, Clark [In Communication Systems and Random Process Theory (Proc. 2nd NATO Advanced Study Inst., Darlington, 1977) (1978) 721–734, Sijthoff & Noordhoff], Clark and Crisan [Probab. Theory Related Fields 133 (2005) 43–56], Davis [Z. Wahrsch. Verw. Gebiete 54 (1980) 125–139], Davis [Teor. Veroyatn. Primen. 27 (1982) 160–167], Kushner [Stochastics 3 (1979) 75–83]. As shown by Davis and Spathopoulos [SIAM J. Control Optim. 25 (1987) 260–278], Davis [In Stochastic Systems: The Mathematics of Filtering and Identification and Applications, Proc. NATO Adv. Study Inst. Les Arcs, Savoie, France 1980 505–528], [In The Oxford Handbook of Nonlinear Filtering (2011) 403–424 Oxford Univ. Press], this type of robust representation is also possible when the signal and the observation noise are correlated, provided the observation process is scalar. For a general correlated noise and multidimensional observations such a representation does not exist. By using the theory of rough paths we provide a solution to this deficiency: the observation process $Y$ is “lifted” to the process $\mathbf{Y}$ that consists of $Y$ and its corresponding Lévy area process, and we show that there exists a continuous map $\theta_{t}^{f}$, defined on a suitably chosen space of Hölder continuous paths such that $\pi_{t}(f)=\theta_{t}^{f}(\mathbf{Y})$, almost surely.

Rough Paths and PDEs

The purpose of the Oberwolfach workshop ”Rough Paths and PDEs” was to bring together these researchers, both young and senior, with the aim to promote progress in rough path theory, the connections with partial differential equations and its applications to numerical methods.

On the rough-paths approach to non-commutative stochastic calculus

We study different possibilities to apply the principles of rough paths theory in a non-commutative probability setting. First, we extend previous results obtained by Capitaine, Donati-Martin and Victoir in Lyonsʼs original formulation of rough paths theory. Then we settle the bases of an alternative non-commutative integration procedure, in the spirit of Gubinelliʼs controlled paths theory, and which allows us to revisit the constructions of Biane and Speicher in the free Brownian case. New approximation results are also derived from the strategy.

On Stratonovich and Skorohod stochastic calculus for Gaussian processes

In this article, we derive a Stratonovich and Skorohod-type change of variables formula for a multidimensional Gaussian process with low Hölder regularity $\gamma$ (typically $\gamma \le1/4$). To this aim, we combine tools from rough paths theory and stochastic analysis.

Compensated fractional derivatives and stochastic evolution equations

We are interested in developing a pathwise theory for mild solutions of stochastic evolution equations when the noise path is β-Hölder continuous for β∈(1/3,1/2). From the point of view of the Rough Path Theory, stochastic integrals related to the solution of ordinary differential equations contain area-elements from a tensor space. Based on (compensated) fractional derivatives we are able to derive a second mild equation for these area components. We formulate sufficient conditions for the existence and uniqueness of a pathwise mild solution by using the Banach fixed point theorem provided that the coefficients of the system are sufficiently regular.

Smoothness of the density for solutions to Gaussian rough differential equations

We consider stochastic differential equations of the form $dY_t=V(Y_t)\,dX_t+V_0(Y_t)\,dt$ driven by a multi-dimensional Gaussian process. Under the assumption that the vector fields $V_0$ and $V=(V_1,\ldots,V_d)$ satisfy H\"{o}rmander's bracket condition, we demonstrate that $Y_t$ admits a smooth density for any $t\in(0,T]$, provided the driving noise satisfies certain nondegeneracy assumptions. Our analysis relies on relies on an interplay of rough path theory, Malliavin calculus and the theory of Gaussian processes. Our result applies to a broad range of examples including fractional Brownian motion with Hurst parameter $H>1/4$, the Ornstein-Uhlenbeck process and the Brownian bridge returning after time $T$.

A SUPPORT THEOREM AND A LARGE DEVIATION PRINCIPLE FOR KUNITA FLOWS

In this paper, stochastic flows driven by Kunita-type stochastic differential equations are studied, focusing on support theorems (ST) and large deviation principles (LDP). We establish a new ST and LDP for Brownian flows with respect to a fine Hölder topology. Our approach is based on recent advances in rough paths theory, which is the natural framework for proving ST and LDP. Nevertheless, while rigorous, our presentation stays rather clear from the rough paths technicalities and is accessible for readers not familiar with them. We view the localized Brownian stochastic flow as a projection of the solution of a rough path differential equation implying the ST and LDP. In a second step the results are generalized for the global flow.

Ordered Forests, Permutations, and Iterated Integrals

We construct an explicit Hopf algebra isomorphism from the algebra of heap-ordered trees to that of quasi-symmetric functions, generated by formal permutations, which is a lift of the natural projection of the Connes-Kreimer algebra of decorated rooted trees onto the shuffle algebra. This isomorphism gives a universal way of lifting measure-indexed characters of the Connes-Kreimer algebra into measure-indexed characters of the shuffle algebra, already introduced in \cite{Unterberger} in the framework of rough path theory as the so-called Fourier normal ordering algorithm.

Regularity of Schramm-Loewner evolutions, annular crossings, and rough path theory

When studying stochastic processes, it is often fruitful to understand several different notions of regularity. One such notion is the optimal Hölder exponent obtainable under reparametrization. In this paper, we show that chordal $\mathrm{SLE}_\kappa$ in the unit disk for $\kappa \le 4$ can be reparametrized to be Hölder continuous of any order up to $1/(1+\kappa/8)$. From this, we obtain that the Young integral is well defined along such $\mathrm{SLE}_\kappa$ paths with probability one, and hence that $\mathrm{SLE}_\kappa$ admits a path-wise notion of integration. This allows us to consider the expected signature of $\mathrm{SLE}$, as defined in rough path theory, and to give a precise formula for its first three gradings. The main technical result required is a uniform bound on the probability that an $\mathrm{SLE}_\kappa$ crosses an annulus $k$-distinct times.