A stochastic calculus is given for processes described by stochastic integrals with respect to fractional Brownian motions and Rosenblatt processes somewhat analogous to the stochastic calculus for Itô processes. These processes for this stochastic calculus arise naturally from a stochastic chain rule for functionals of Rosenblatt processes; and some Itô-type expressions are given here. Furthermore, there is some analysis of these results for their applications to problems using Rosenblatt noise.
We analyze {\em the Rosenblatt process} which is a selfsimilar process with stationary increments and which appears as limit in the so-called {\em Non Central Limit Theorem} (Dobrushin and Major (1979), Taqqu (1979)). This process is non-Gaussian and it lives in the second Wiener chaos. We give its representation as a Wiener-Itô multiple integral with respect to the Brownian motion on a finite interval and we develop a stochastic calculus with respect to it by using both pathwise type calculus and Malliavin calculus.
In this paper, we define a stochastic calculus with respect to the Rosenblatt process by means of white noise distribution theory. For this purpose, we compute the translated characteristic function of the Rosenblatt process at time $t>0$ in any direction $ξ\in S(\mathbb{R})$ and the derivative of the Rosenblatt process in the white noise sense. Using Wick multiplication by the former derivative and Pettis integration, we define our stochastic integral with respect to the Rosenblatt process for a wide class of distribution processes. We obtain an explicit formula for the variance of such a sto…
We use techniques of Malliavin calculus to study the convergence in law of a family of generalized Rosenblatt processes $Z_γ$ with kernels defined by parameters $γ$ taking values in a tetrahedral region $Δ$ of $\RR^q$. We prove that, as $γ$ converges to a face of $Δ$, the process $Z_γ$ converges to a compound Gaussian distribution with random variance given by the square of a Rosenblatt process of one lower rank. The convergence in law is shown to be stable. This work generalizes a previous result of Bai and Taqqu, who proved the result in the case $q=2$ and without stability.
We study the least squares estimator for the drift parameter of the Langevin stochastic equation driven by the Rosenblatt process. Using the techniques of the Malliavin calculus and the stochastic integration with respect to the Rosenblatt process, we analyze the consistency and the asymptotic distribution of this estimator. We also introduce alternative estimators, which can be simulated, and we study their asymptotic properties.
Using multiple stochastic integrals and the Malliavin calculus, we analyze the asymptotic behavior of quadratic variations for a specific non-Gaussian self-similar process, the Rosenblatt process. We apply our results to the design of strongly consistent statistical estimators for the self-similarity parameter $H$. Although, in the case of the Rosenblatt process, our estimator has non-Gaussian asymptotics for all $H>1/2$, we show the remarkable fact that the process's data at time 1 can be used to construct a distinct, compensated estimator with Gaussian asymptotics for $H\in(1/2,2/3)$.
By using chaos expansion into multiple stochastic integrals, we make a wavelet analysis of two self-similar stochastic processes: the fractional Brownian motion and the Rosenblatt process. We study the asymptotic behavior of the statistic based on the wavelet coefficients of these processes. Basically, when applied to a non-Gaussian process (such as the Rosenblatt process) this statistic satisfies a non-central limit theorem even when we increase the number of vanishing moments of the wavelet function. We apply our limit theorems to construct estimators for the self-similarity index and we ill…
The most known example of a class of non-Gaussian stochastic processes which belongs to the homogenous Wiener chaos of an arbitrary order N > 1 are probably Hermite processes of rank N. They generalize fractional Brownian motion (fBm) and Rosenblatt process in a natural way. They were introduced several decades ago. Yet, in contrast with fBm and many other Gaussian and stable stochastic processes and fields related to it, few results on path behavior of Hermite processes are available in the literature. For instance the natural issue of whether or not their paths are nowhere differentiable fun…
This work is about a new class of martingales: the vertical martingales. We construct the vertical martingale for smooth submersions and we develop a stochastic calculus for one. Furthermore, we gives a stochastic characterization for harmonic sections.
We compute Greeks for stochastic volatility models driven by Brownian informations. We use the Malliavin method introduced for deterministic volatility models.
The integration-by-parts formula discovered by Malliavin for the Ito map on Wiener space is proved using the two-parameter stochastic calculus. It is also shown that the solution of a one-parameter stochastic differential equation driven by a two-parameter semimartingale is itself a two-parameter semimartingale.
This paper first summarizes the foundations of stochastic calculus via regularization and constructs through this procedure Itô and Stratonovich integrals. In the second part, a survey and new results are presented in relation with finite quadratic variation processes, Dirichlet and weak Dirichlet processes.
This chapter is divided into two parts. The first is largely expository and builds on Karandikar's axiomatisation of It{ô} calculus for matrix-valued semimartin-gales. Its aim is to unfold in detail the algebraic structures implied for iterated It{ô} and Stratonovich integrals. These constructions generalise the classical rules of Chen calculus for deterministic scalar-valued iterated integrals. The second part develops the stochastic analog of what is commonly called chronological calculus in control theory. We obtain in particular a pre-Lie Magnus formula for the logarithm of the It{ô} stoch…
This paper begins by giving an historical context to fractional Brownian Motion and its development. Section 2 then introduces the fractional calculus, from the Riemann-Liouville perspective. In Section 3, we introduce Brownian motion and its properties, which is the framework for deriving the Itô integral. In Section 4 we finally introduce the Itô calculus and discuss the derivation of the Itô integral. Section 4.1 continues the discussion about the Itô calculus by introducing the Itô formula, which is the analogue to the chain rule in classical calculus. In Section 5 we present our formal de…
In this paper, we study rough path properties of stochastic integrals of Itô's type and Stratonovich's type with respect to $G$-Brownian motion. The roughness of $G$-Brownian Motion is estimated and then the pathwise Norris lemma in $G$-framework is obtained.
Several versions of Itô's formula have been obtained in the context of the functional stochastic calculus. Here, we revisit this topic in two ways. First, by defining a notion of derivative along a functional, we extend the setting of the (semimartingale) functional Itô's formula and corresponding calculus. Second, for Lévy processes, an optimal local-time based Itô's formula is obtained. Some quick applications are then given.
In this book, we introduce a new approach of sublinear expectation to deal with the problem of probability and distribution model uncertainty. We a new type of (robust) normal distributions and the related central limit theorem under sublinear expectation. We also present a new type of Brownian motion under sublinear expectations and the related stochastic calculus of Ito's type. The results provide robust tools for the problem of probability model uncertainty arising from financial risk management, statistics and stochastic controls.
Malliavin Calculus can be seen as a differential calculus on Wiener spaces. We present the notion of stochastic manifold for which the Malliavin Calculus plays the same role as the classical differential calculus for the differential manifolds. The set of the paths in a Riemmanian compact manifold is then seen as a particular case of the above structure.
We give a stochastic calculus proof of the Central Limit Theorem \[ {\int (L^{x+h}_{t}- L^{x}_{t})^{2} dx- 4ht\over h^{3/2}} \stackrel{\mathcal{L}}{\Longrightarrow}c(\int (L^{x}_{t})^{2} dx)^{1/2} η\] as $h\to 0$ for Brownian local time $L^{x}_{t}$. Here $η$ is an independent normal random variable with mean zero and variance one.
Nakao's stochastic integrals for continuous additive functionals of zero energy are extended from the symmetric Dirichlet forms setting to the non-symmetric Dirichlet forms setting. Ito's formula in terms of the extended stochastic integrals is obtained.
Stochastic integration with respect to Gaussian processes, such as fractional Brownian motion (fBm) or multifractional Brownian motion (mBm), has raised strong interest in recent years, motivated in particular by applications in finance, Internet traffic modeling and biomedicine. The aim of this work to define and develop, using White Noise Theory, an anticipative stochastic calculus with respect to a large class of Gaussian processes, denoted G, that contains, among many other processes, Volterra processes (and thus fBm) and also mBm. This stochastic calculus includes a definition of a stocha…
We describe stochastic calculus in the context of processes that are driven by an adapted point process of locally finite intensity and are differentiable between jumps. This includes Markov chains as well as non-Markov processes. By analogy with Itô processes we define the drift and diffusivity, which we then use to describe a general sample path estimate. We then give several examples, including ODE approximation, processes with linear drift, first passage times, and an application to the stochastic logistic model.
We consider Riemann sum approximations of stochastic integrals with respect to the fractional Browian motion of index $H\geq \frac12$. We show the convergence of these schemes at first and second order. The processes obtained in the limit in the second case are stochastic integrals with respect to the Rosenblatt process if $H >\frac34$ and the standard Brownian motion otherwise. These results are obtained under the assumption that the integrand is a `controlled' process. We provide many examples of such processes, in particular fractional semimartingales and multiple Wiener-Itô integrals
In this article, we derive a Stratonovich and Skorohod type change of variables formula for a multidimensional Gaussian process with low Hölder regularity (typically lower than 1/4). To this aim, we combine tools from rough paths theory and stochastic analysis.
We obtain sharp sufficient conditions for exponentially integrable stochastic processes $X=\{X(t)\!\!: t\in [0,1]\}$, to have sample paths with bounded $Φ$-variation. When $X$ is moreover Gaussian, we also provide a bound of the expectation of the associated $Φ$-variation norm of $X$. For an Hermite process $X$ of order $m\in \N$ and of Hurst index $H\in (1/2,1)$, we show that $X$ is of bounded $Φ$-variation where $Φ(x)=x^{1/H}(\log(\log 1/x))^{-m/(2H)}$, and that this $Φ$ is optimal. This shows that in terms of $Φ$-variation, the Rosenblatt process (corresponding to $m=2$) has more rough samp…
We develop a notion of nonlinear expectation --G-expectation-- generated by a nonlinear heat equation with infinitesimal generator G. We first study multi-dimensional G-normal distributions. With this nonlinear distribution we can introduce our G-expectation under which the canonical process is a multi dimensional G-Brownian motion. We then establish the related stochastic calculus, especially stochastic integrals of Ito's type with respect to our G-Brownian motion and derive the related Ito's formula. We have also obtained the existence and uniqueness of stochastic differential equation under…
We consider a stationary sequence $(X_n)$ constructed by a multiple stochastic integral and an infinite-measure conservative dynamical system. The random measure defining the multiple integral is non-Gaussian, infinitely divisible and has a finite variance. Some additional assumptions on the dynamical system give rise to a parameter $β\in(0,1)$ quantifying the conservativity of the system. This parameter $β$ together with the order of the integral determines the decay rate of the covariance of $(X_n)$. The goal of the paper is to establish limit theorems for the partial sum process of $(X_n)$.…
Stochastic integration \textit{wrt} Gaussian processes has raised strong interest in recent years, motivated in particular by its applications in Internet traffic modeling, biomedicine and finance. The aim of this work is to define and develop a White Noise Theory-based anticipative stochastic calculus with respect to all Gaussian processes that have an integral representation over a real (maybe infinite) interval. Very rich, this class of Gaussian processes contains, among many others, Volterra processes (and thus fractional Brownian motion) as well as processes the regularity of which varies…
By using Malliavin calculus and multiple Wiener-Itô integrals, we study the existence and the regularity of stochastic currents defined as Skorohod (divergence) integrals with respect to the Brownian motion and to the fractional Brownian motion. We consider also the multidimensional multiparameter case and we compare the regularity of the current as a distribution in negative Sobolev spaces with its regularity in Watanabe space.
A *-algebraic indefinite structure of quantum stochastic (QS) calculus is introduced and a continuity property of generalized nonadapted QS integrals is proved under the natural integrability conditions in an infinitely dimensional nuclear space. The class of nondemolition output QS processes in quantum open systems is characterized in terms of the QS calculus, and the problem of QS nonlinear filtering with respect to nondemolition continuous measurments is investigated. The stochastic calculus of a posteriori conditional expectations in quantum observed systems is developed and a general quan…
We give a new characterization for the convergence in distribution to a standard normal law of a sequence of multiple stochastic integrals of a fixed order with variance one, in terms of the Malliavin derivatives of the sequence. We extend our result to the multidimensional case and prove a weak convergence result for a sequence of square integrable random variables.
In the first part of this paper I give the historical background to my initial interest in stochastic analysis and to the writing of my book Stochastic Differential Equations. The first edition of this book was published by Springer in 1985, with the highly appreciated support of Catriona Byrne. In the second part I present a motivation for modelling the dynamics of a system subject to a noise by means of a stochastic partial differential equation (SPDE) driven by a time-space Brownian sheet. This is followed by a brief survey of time-space white noise and Hida-Malliavin calculus, which are us…
We provide a simple proof, as well as several generalizations, of a recent result by Davis and Suh, characterizing a class of continuous submartingales and supermartingales that can be expressed in terms of a squared Brownian motion and of some appropriate powers of its maximum. Our techniques involve elementary stochastic calculus, as well as the Doob-Meyer decomposition of continuous submartingales. These results can be used to obtain an explicit expression of the constants appearing in the Burkholder-Davis-Gundy inequalities. A connection with some balayage formulae is also established.
Controlled fields, rough stochastic calculus, and Itô-Wentzell-Alekseev-Gröbner identities
Technische Universität Berlin（柏林工业大学）；Weierstraß Institut（魏尔斯特拉斯研究所）；The Chinese University of Hong Kong, Shenzhen (CUHK-Shenzhen)（香港中文大学（深圳））；University of Münster（明斯特大学）；Research Center for Mathematics and Interdisciplinary Sciences, Shandong University（山东大学数学与交叉科学研究中心）；Frontiers Science Center for Nonlinear Expectations, Ministry of Education, Shandong University（山东大学非线性期望前沿科学中心）
In this note we show how to generalize the stochastic calculus method introduced by Comets and Neveu [Comm. Math. Phys. 166 (1995) 549-564] for two models of spin glasses, namely, the SK model with external field and the perceptron model. This method allows to derive quite easily some fluctuation results for the free energy in those two cases.
Using time-reversal, we introduce a stochastic integral for zero-energy additive functionals of symmetric Markov processes, extending earlier work of S. Nakao. Various properties of such stochastic integrals are discussed and an Itô formula for Dirichlet processes is obtained.
T. Lyons' rough path theory is something like a deterministic version of K. Ito's theory of stochastic differential equations, combined with ideas from K. T. Chen's theory of iterated path integrals. In this article we survey rough path theory, in particular, its probabilistic aspects.
We refine stochastic calculus for symmetric Markov processes without using time reverse operators. Under some conditions on the jump functions of locally square integrable martingale additive functionals, we extend Nakao's divergence-like continuous additive functional of zero energy and the stochastic integral with respect to it under the law for quasi-everywhere starting points, which are refinements of the previous results under the law for almost everywhere starting points. This refinement of stochastic calculus enables us to establish a generalized Fukushima decomposition for a certain cl…
We introduce a notion of nonlinear expectation --G--expectation-- generated by a nonlinear heat equation with infinitesimal generator G. We first discuss the notion of G-standard normal distribution. With this nonlinear distribution we can introduce our G-expectation under which the canonical process is a G--Brownian motion. We then establish the related stochastic calculus, especially stochastic integrals of Ito's type with respect to our G--Brownian motion and derive the related Ito's formula. We have also give the existence and uniqueness of stochastic differential equation under our G-expe…
We develop a stochastic calculus for processes which are built by convoluting a pure jump, zero expectation Lévy process with a Volterra-type kernel. This class of processes contains, for example, fractional Lévy processes as studied by Marquardt [Bernoulli 12 (2006) 1090--1126.] The integral which we introduce is a Skorokhod integral. Nonetheless, we avoid the technicalities from Malliavin calculus and white noise analysis and give an elementary definition based on expectations under change of measure. As a main result, we derive an Itô formula which separates the different contributions from…
We consider the class of all the Hermite processes $(Z_{t}^{(q,H)})_{t\in \lbrack 0,1]}$ of order $q\in \mathbf{N}^{\ast}$ and with Hurst parameter $% H\in (\frac{1}{2},1)$. The process $Z^{(q,H)}$ is $H$-selfsimilar, it has stationary increments and it exhibits long-range dependence identical to that of fractional Brownian motion (fBm). For $q=1$, $Z^{(1,H)}$ is fBm, which is Gaussian; for $q=2$, $Z^{(2,H)}$ is the Rosenblatt process, which lives in the second Wiener chaos; for any $q>2$, $Z^{(q,H)}$ is a process in the $q$th Wiener chaos. We study the variations of $Z^{(q,H)}$ for any $q$, b…
This article deals with classes of antagonistic games with two players. A game is specified in terms of two `hostile' stochastic processes representing mutual attacks upon random times exerting casualties of random magnitudes. The game ends when one of the players is defeated. We target the first passage time $τ_ρ$ of the defeat and the number of casualties to either player upon $τ_ρ$. Here we validate our claim of analytic tractability of the general formulas obtained in [1] under various transforms.
We consider anticipative Stratonovich stochastic differential equations driven by some stochastic process (not necessarily a semi-martingale). No adaptedness of initial point or vector fields is assumed. Under a simple condition on the stochastic process, we show that the unique solution of the above SDE understood in the rough path sense is actually a Stratonovich solution. This condition is satisfied by the Brownian motion and the fractional Brownian motion with Hurst parameter greater than 1/4. As application, we obtain rather flexible results such as support theorems, large deviation princ…
It is shown that under a certain condition on a semimartingale and a time-change, any stochastic integral driven by the time-changed semimartingale is a time-changed stochastic integral driven by the original semimartingale. As a direct consequence, a specialized form of the Ito formula is derived. When a standard Brownian motion is the original semimartingale, classical Ito stochastic differential equations driven by the Brownian motion with drift extend to a larger class of stochastic differential equations involving a time-change with continuous paths. A form of the general solution of line…
Functional Itô calculus was introduced in order to expand a functional $F(t, X\_{\cdot+t}, X\_t)$ depending on time $t$, past and present values of the process $X$. Another possibility to expand $F(t, X\_{\cdot+t}, X\_t)$ consists in considering the path $X\_{\cdot+t}=\{X\_{x+t},\,x\in[-T,0]\}$ as an element of the Banach space of continuous functions on $C([-T,0])$ and to use Banach space stochastic calculus. The aim of this paper is threefold. 1) To reformulate functional Itô calculus, separating time and past, making use of the regularization procedures which matches more naturally the noti…
We introduce the notion of {\em covariance measure structure} for square integrable stochastic processes. We define Wiener integral, we develop a suitable formalism for stochastic calculus of variations and we make Gaussian assumptions only when necessary. Our main examples are finite quadratric variation processes with stationary increments and the bifractional Brownian motion.
In this paper we study backward stochastic differential equations with general terminal value and general random generator. In particular, we do not require the terminal value be given by a forward diffusion equation. The randomness of the generator does not need to be from a forward equation, either. Motivated from applications to numerical simulations, first we obtain the $L^p$-Hölder continuity of the solution. Then we construct several numerical approximation schemes for backward stochastic differential equations and obtain the rate of convergence of the schemes based on the obtained $L^p$…
Contributions of the present paper consist of two parts. In the first one, we contribute to the theory of stochastic calculus for signed measures. For instance, we provide some results permitting to characterize martingales and Brownian motion both defined under a signed measure. We also prove that the uniformly integrable martingales (defined with respect to a signed measure) can be expressed as relative martingales and we provide some new results to the study of the class $Σ(H)$ which appeared for the first time in \cite{f} and studied in \cite{f,e,o}. The second part is devoted to the const…
The stochastic partial differential equation analyzed in this work, is motivated by a simplified mesoscopic physical model for phase separation. It describes pattern formation due to adsorption and desorption mechanisms involved in surface processes, in the presence of a stochastic driving force. This equation is a combination of Cahn-Hilliard and Allen-Cahn type operators with a multiplicative, white, space-time noise of unbounded diffusion. We apply Malliavin calculus, in order to investigate the existence of a density for the stochastic solution $u$. In dimension one, according to the regul…
We study Malliavin differentiability of solutions to sub-critical singular parabolic stochastic partial differential equations (SPDEs) and we prove the existence of densities for a class of singular SPDEs. Both of these results are implemented in the setting of regularity structures. For this we construct renormalized models in situations where some of the driving noises are replaced by deterministic Cameron-Martin functions, and we show Lipschitz continuity of these models with respect to the Cameron-Martin norm. In particular, in many interesting situations we obtain a convergence and stabil…
A derivation operator and a divergence operator are defined on the algebra of bounded operators on the symmetric Fock space over the complexification of a real Hilbert space $\eufrak{h}$ and it is shown that they satisfy similar properties as the derivation and divergence operator on the Wiener space over $\eufrak{h}$. The derivation operator is then used to give sufficient conditions for the existence of smooth Wigner densities for pairs of operators satisfying the canonical commutation relations. For $\eufrak{h}=L^2(\mathbb{R}_+)$, the divergence operator is shown to coincide with the Hudson…
