Cylindrical continuous martingales and stochastic integration in infinite dimensions

The aim of this paper is to investigate the almost sure stability with a certain rate function λ(t) for a class of stochastic evolution equations in infinite dimensional spaces under various sufficient conditions. The results obtained here include exponential and polynomial stability as special cases. Much more refined sufficient conditions than the usual ones, for example, those described in [14], are obtained under our framework by the method of Liapunov functions. Two examples are given to illustrate our theory.

Recently, van Neerven, Weis and the author, constructed a theory for stochastic integration of UMD Banach space valued processes. Here the authors use a (cylindrical) Brownian motion as an integrator. In this note we show how one can extend these results to the case where the integrator is an arbitrary real-valued continuous local martingale. We give several characterizations of integrability and prove a version of the Itô isometry, the Burkholder–Davis–Gundy inequality, the Itô formula and the martingale representation theorem.

In this paper, we consider Meyer–Yoeurp decompositions for UMD Banach space-valued martingales. Namely, we prove that $X$ is a UMD Banach space if and only if for any fixed $p\in(1,\infty)$, any $X$-valued $L^{p}$-martingale $M$ has a unique decomposition $M=M^{d}+M^{c}$ such that $M^{d}$ is a purely discontinuous martingale, $M^{c}$ is a continuous martingale, $M^{c}_{0}=0$ and \[\mathbb{E}\big\|M^{d}_{\infty}\big\|^{p}+\mathbb{E}\big\|M^{c}_{\infty}\big\|^{p}\leq c_{p,X}\mathbb{E}\|M_{\infty}\|^{p}.\] An analogous assertion is shown for the Yoeurp decomposition of a purely discontinuous martingales into a sum of a quasi-left continuous martingale and a martingale with accessible jumps. As an application, we show that $X$ is a UMD Banach space if and only if for any fixed $p\in(1,\infty)$ and for all $X$-valued martingales $M$ and $N$ such that $N$ is weakly differentially subordinated to $M$, one has the estimate $\mathbb{E}\|N_{\infty}\|^{p}\leq C_{p,X}\mathbb{E}\|M_{\infty}\|^{p}$.

We present a new test for the “continuous martingale hypothesis”. That is, a test for the hypothesis that observed data are from a process which is a continuous local martingale. The basis of the test is an embedded random walk at first passage times, obtained from the well-known representation of a continuous local martingale as a continuous time-change of Brownian motion. With a variety of simulated diffusion processes our new test shows higher power than existing tests using either the crossing tree or the quadratic variation, including the situation where non-negligible drift is present. The power of the test in the presence of jumps is also explored with a variety of simulated jump diffusion processes. The test is also applied to two sequences of high-frequency foreign exchange trade-by-trade data. In both cases the continuous martingale hypothesis is rejected at times less than hourly and we identify significant dependence in price movements at these small scales.

The main purpose of this article is to propose a reasonable definition for the stochastic integration (S.I.) of progressive processes w.r.t. semimartingales. This S.I. generalizes that of predictable processes w.r.t. semimartingales as well as the stochastic Stieltjes integration. This S.I. is proposed in §1. We give also in §1 an exponential formula for semimartingales using this S.I.. The rest of this paper consists of several remarks on the theory of stochastic integration which are mostly of pedagogical interest. In §2 we propose a new construction of the S.I. of predictable processes w.r.t. local martingales. A simple proof of the integration by parts formula is given in §3. Finally, we propose in §4 a short proof of Meyer’s theorem on compensated stochastic integrals of local martingales.AMS Subject Classification60H05

This thesis consists of two parts, each part concentrating on a different problem from the theory of Stochastic Integration. Chapter 1 contains the introduction explaining the results in this dissertation in general terms. We use the infinite dimensional space S'R endowed with the gaussian measure mu. The Hilbert space ( L2) is defined as (L2) (L2) ≡ L2( S'R , mu) and our results are based on the Gel'fand triple ( S )beta ⊂ (L2) ⊂ S* b . The necessary preliminary background in white noise analysis are well elaborated in Chapter 2. In Chapter 3 we present a generalization of the Ito Formula to anticipating processes in the white noise framework. We first introduce an extension of the Ito integral to anticipating processes called the Hitsuda Skorokhod integral. We then use the anticipating Ito formula for processes of the type theta(X(t), B(c)), where X(t) is a Wiener integral and B(t) = 〈·, 1&sqbl0;0,t&parr0; 〉 is Brownian motion for a ≤ c ≤ b, which was obtained in Professor H. H. Kuo's book, White Noise Distribution Theory, as a motivation to obtain our first main result. We generalize the formula in Prof. Kuo's book to processes of the form theta(X(t),F) where X(t) is a Wiener integral and F ∈ W 1/2, a special subspace of (L 2). Chapter 4 contains the second part of our work. We first state the Clark-Ocone (C-O) formula in the white noise framework. We then verify the formula for Brownian functionals using the very important white noise tool, the S-transform. As our second main result, we extend the formula to generalized Brownian functionals of the form F=Sinfinityn=0 :˙⊗n:,fn where fn ∈ L 2 R ⊗&d4;n . As examples we verify the formula for the Hermite Brownian functional: B(t)n:t using the Ito formula. We also compute the C-O formula for the Donsker's delta function using the result for the Hermite Brownian functional. Finally, the formula is generalized to compositions of tempered distributions with Brownian motion, i.e, to functionals of the form f( B(t)) where f ∈ S'R and B(t) = B( t) = 〈·, 1&sqbl0;0,t&parr0; 〉 is Brownian motion.

This chapter is devoted to stochastic evolution equations in Hilbert spaces, in particular stochastic parabolic type equations of the form $$\displaystyle{du(t,x,\omega ) = \mathcal{L}(t,\omega )u(t,x,\omega )\,dt + \mathcal{M}(t,\omega )u(t,x,\omega )\,dW_{Q}(t),}$$ where $$\mathcal{L}$$ and $$\mathcal{M}$$ are second-order elliptic and first-order differential operators and W Q is a colored Wiener process (see Example 5.1). These equations are generalization of finite-dimensional SDEs and appear in the study of random phenomena in natural sciences and the unnormalized conditional probability of finite-dimensional diffusion processes (see Sect. 5.5), related to filtering equations derived in Fujisaki et al. (Osaka J Math 9:19–40, 1972) and Kushner (J Differ Equ 3:179–190, 1967). In Sect. 5.1 we collect basic definitions and results for Hilbert space-valued processes; in particular, for continuous martingales, quadratic variations and correlation operators are treated. Stochastic integrals are introduced in Sect. 5.2. Section 5.3 is devoted to the study of stochastic parabolic equations from the viewpoint of Hilbert space-valued SDEs, following Rozovskii (Stochastic evolution systems. Kluwer Academic, Dordrecht/Boston, 1990). By using the results presented, we also consider a semilinear stochastic parabolic equation with Lipschitz nonlinearity in Sect. 5.3.4. Section 5.4 deals with Itô’s formula and in Sect. 5.5 Zakai equations related to filtering problems are given.

Stochastic evolution equations in UMD Banach spaces

This chapter is a self-contained introduction to the theory of stochastic integration with respect to continuous semimartingales. Apart from some simple facts on the predictable σ-field (which are included with proofs in Section 1.2), we do not need any other result from the general theory of processes. The Doob—Meyer decomposition of the square of a continuous (local) martingale is given by constructing the quadratic variation process directly. Once this is done, the stochastic integral is defined and its properties are obtained in a natural fashion, using arguments common in the theory of the Lebesgue integral, namely, the monotone class theorem and dominated convergence theorem. We first give the definition of the stochastic integral with respect to Brownian motion and then deal with the case of continuous semimartingales in stages. In the last section, we define the integral with respect to a general (r.c.l.l.) semimartingale. However, this time we state without proof some key results, including the Doob—Meyer decomposition. We show that these results, three to be precise, enable us to define the integral in the general case and derive the properties of the integral following the route adopted earlier.KeywordsBrownian MotionDominate Convergence TheoremQuadratic VariationStochastic IntegrationLocal MartingaleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We prove pathwise (hence strong) uniqueness of solutions to stochastic evolution equations in Hilbert spaces with merely measurable bounded drift and cylindrical Wiener noise, thus generalizing Veretennikov's fundamental result on $\mathbb{R}^d$ to infinite dimensions. Because Sobolev regularity results implying continuity or smoothness of functions do not hold on infinite-dimensional spaces, we employ methods and results developed in the study of Malliavin-Sobolev spaces in infinite dimensions. The price we pay is that we can prove uniqueness for a large class, but not for every initial distribution. Such restriction, however, is common in infinite dimensions.

We establish weak convergence rates for noise discretizations of a wide class of stochastic evolution equations with non-regularizing semigroups and additive or multiplicative noise. This class covers the nonlinear stochastic wave, HJMM, stochastic Schr\"odinger and linearized stochastic Korteweg-de Vries equation. For several important equations, including the stochastic wave equation, previous methods give only suboptimal rates, whereas our rates are essentially sharp.

: This paper is the second in a series by the authors concerned with the theory of viscosity solutions Hamilton-Jacobi equations in infinite dimensional spaces. The first paper introduced a notion of viscosity solution appropriate for the study of Hamilton-Jacobi equations in spaces with the so-called Radon-Nikodym property and obtained uniqueness theorems under assumptions paralleling the finite dimensional theory. The main results of the current paper concern existence of solutions of stationary and time-dependent Hamilton-Jacobi equations. In order to establish these results it is necessary to overcome the difficulties associated with the fact that bounded sets are not precompact in infinite dimensions and this is done by sharp constructive estimates coupled with the use of differential games to solve regularized problems. Interest in this subject arises on the abstract side from the desire to contribute to the theory of linear partial differential equations in infinite dimensional spaces to treat natural questions raised by the finite dimensional theory. Interest also arises from potential applications to the theory of control of partial differential equations. However, the results herein do not apply directly to problems of the form arising in the control of partial differential equations, a question which wil be treated in the next paper of the series. Additional keywords: Banach spaces, Existence theory. (Author)

A class of stochastic evolution equations with additive noise and weakly continuous drift is considered. First, regularity properties of the corresponding Ornstein-Uhlenbeck transition semigroupRt are obtained. We show thatRt is a compactC0-semigroup in all Sobolev spacesWn,p which are built on its invariant measure μ. Then we show the existence, uniqueness, compactness and smoothing properties of the transition semigroup for semilinear equations inLp(μ) spaces and spacesW1,p. As a consequence we prove the uniquencess of martingale solutions to the stochastic equation and the existence of a unique invariant measure equivalent to μ. It is shown also that the density of this measure with respect to μ is inLp(μ) for allp≧1.

The purpose of this dissertation is to establish, in a certain infinite dimensional setting, some existence and uniqueness results pertaining to the type of exponential martingale problem previously considered by Strook and Varadhan in the finite dimensional setting. The setting is among those considered by K. Ito in his work on stochastic differential equations in infinite dimensional spaces. Briefly, the martingale problem takes the following form: to show the existence and uniqueness of a measure m on the space S of continuous functions defined on the nonnegative real lien with values in some state space, having the property that exp(f(t,w)) is an m-martingale, f being a function of type, with t denoting a time parameter, and w denoting a generic element of S. The measure m is then called a solution of the martingale problem. In this dissertation, the state spaces under consideration are subspaces of the space of tempered distributions on the real line. The existence of solutions is established under certain continuity conditions by constructing a family of measures satisfying a Prokhorov criterion for relative compactness with respect to weak convergence, and then showing that a weak limit point of that family has the desired martingale property. Secondly, under somewhat stronger assumptions, a uniqueness result for a previously constructed solution is proved in a manner similar to that in which the uniqueness of a related stochastic differential equation is established. Finally, the relation of the martingale problem to certain infinite dimensional stochastic integrals and stochastic equations is pointed out.

We study a class of stochastic evolution equations with a dissipative forcing nonlinearity and additive noise. The noise is assumed to satisfy rather general assumptions about the form of the covariance function; our framework covers examples of Gaussian processes, like fractional and bifractional Brownian motion and also non Gaussian examples like the Hermite process. We give an application of our results to the study of the stochastic version of a common model of potential spread in a dendritic tree. Our investigation is specially motivated by possibility to introduce long-range dependence in time of the stochastic perturbation.