On intensities of perturbed random measures on Hausdorff spaces

Abstract

This article studies classes of random measures on topological spaces perturbed by stochastic processes (a.k.a. modulated random measures). We render a rigorous construction of the stochastic integral of functions of two variables and showed that such an integral is a random measure. We establish a new Campbell-type formula that, along with a rigorous construction of modulation, leads to the intensity of a modulated random measure. Mathematical formalism of integral-driven random measures and their stochastic intensities find numerous applications in stochastic models, physics, astrophysics, and finance that we discuss throughout the article.     

On an approach to the definition of a stochastic integral with respect to a Gaussian measure

We define a stochastic Riemann integral with respect to a Gaussian measure. The class of integrable functions is introduced in which there exists a solution of a stochastic Fredholm integral equation. It is shown by examples how to pass from the integral defined here to the Itô and Stratonovich integrals.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 100–108, 1986.     

Stochastic integration on partially ordered sets

The stochastic integral is introduced with respect to a stochastic process X = (X_s)_sεV, where V is any general partially ordered set satisfying some mild regularity conditions. As important examples the stochastic integral is constructed with respect to a class of Gaussian processes having similarities to the Brownian motion on the real line, and also with respect to L²-martingales under an assumption of conditional independence on the underlying σ-fields.     

Stochastic integral with respect to a semi-Markov process of diffusion type

We consider a multidimensional semi-Markov process of diffusion type. A stochastic integral with respect to the semi-Markov process is defined in terms of asymptotics related to the first exit time from a small neighborhood of the starting point of the process, and, in particular, in terms of its characteristic operator. This integral is equal to the sum of two other integrals: the first one is a curvilinear integral with respect to an additive functional defined in terms of the expected first exit time from a small neighborhood, and the second one is a stochastic integral with respect to a martingale of special kind. To prove the existence and to derive the properties of the integral, both the method of deducing sequences and that of inscribed ellipsoids are used. For Markov processes of diffusion type, the new definition of the stochastic integral is reduced to the standard one. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 328, 2005, pp. 251–276.     

Central limit theorem for an iterated integral with respect to fBm with H>1/2

We construct an iterated stochastic integral with respect to fractional Brownian motion (fBm) with H>1/2. The first integrand is a deterministic function, and each successive integral is with respect to an independent fBm. We show that this symmetric stochastic integral is equal to the Malliavin divergence integral. By a version of the Fourth Moment Theorem of Nualart and Peccati [10], we show that a family of such integrals converges in distribution to a scaled Brownian motion. An application is an approximation to the windings for a planar fBm, previously studied by Baudoin and Nualart [2].     

Existence of Random Solutions of a General Class of Stochastic Functional Integral Equations

Abstract

In this paper, we establish the existence of solutions of a more general class of stochastic functional integral equations. The main tools here are the measure of noncompactness and the fixed point theorem of Darbo type. The results of this paper generalize the results of Rao–Tsokos [Rao, A.N.V.; Tsokos, C.P. A class of stochastic functional integral equations. Coll. Math. 1976, 35, 141–146.] and Szynal–Wedrychowicz [Szynal, D.; Wedrychowicz, S. On existence and an asymptotic behaviour of random solutions of a class of stochastic functional integral equations. Coll. Math. 1987, 51, 349–364.].     

Convergence of delay differential equations driven by fractional Brownian motion

In this note, we prove an existence and uniqueness result of solution for stochastic differential delay equations with hereditary drift driven by a fractional Brownian motion with Hurst parameter H > 1/2. Then, we show that, when the delay goes to zero, the solutions to these equations converge, almost surely and in L ^p , to the solution for the equation without delay. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann–Stieltjes integral.     

On the approximation of the solutions of stochastic equations with θ-integrals

The paper investigates the problem of approximation of stochastic θ-integrals and the solutions of stochastic differential equations. The complete classification of the methods of approximation of stochastic θ-integrals in the convolution algebra is proposed. It is proved that the solutions of stochastic integral equations with θ-integral can be approximated by the solutions of finite-difference equations with averaging.     

Stochastic integral of divergence type with respect to fractional Brownian motion with Hurst parameter

We define a stochastic integral with respect to fractional Brownian motion B^H with Hurst parameter that extends the divergence integral from Malliavin calculus. For this extended divergence integral we prove a Fubini theorem and establish versions of the formulas of Itô and Tanaka that hold for all . Then we use the extended divergence integral to show that for every and all , the Russo–Vallois symmetric integral exists and is equal to , where G^′=g, while for , does not exist.     

Nonlinear stochastic differential equations in infinite dimensions

The stochastic variation of constants proved in[2] results to be an interesting tool to study properties of different classes of stochastic differential equations. In particular, we study the extension to the case of coefficients depending on the solution X _t. It turns out that the representation formula becomes a stochastic integral equation that has to be studied via anticipate calculas     

Optimal control of certain hyperbolic and integral stochastic equations

Using Gateaux differentiation of the quality functional we obtain necessary conditions for optimality of a control for stochastic differential equations of hyperbolic type containing two-parameter white noise and for stochastic integral equations.Translated fromTeoriya Sluchainykh Protsessov, Vol. 15, pp. 110–116, 1987.     

On tightness of solutions of stochastic integral equations driven by <Emphasis Type="Italic">p</Emphasis>-semimartingales

We obtain sufficient conditions for tightness of a sequence of solutions of stochastic integral equations driven by p-semimartingales.     

Stochastic Integrals of Continuous Local Martingales,I

Consider a continuous local martingale X. We say that X satisfies the representation property if any martingale Y of X can be represented as stochastic ITǒ integral of X. Using the method of random time change systematically, in the present paper the representation problem for continuous local martingales is treated. We describe a class of martingales Y that can be represented as stochastic integral of X by probabilistic conditions. This leads to sufficient conditions for the representation property of X being true. Besides, an interesting characterization of continuous processes with independent increments is obtained. In part II. we proceed with general examples, applications to the n-dimensional case, and, in particular, to the n-dimensional time change of continuous local martingales with orthogonal components.     

Existence and exponential stability for neutral stochastic fractional differential equations with impulses driven by Poisson jumps

The paper is mainly concerned with a class of neutral stochastic fractional integro-differential equation with Poisson jumps. First, the existence and uniqueness for mild solution of an impulsive stochastic system driven by Poisson jumps is established by using the Banach fixed point theorem and resolvent operator. The exponential stability in the pth moment for mild solution to neutral stochastic fractional integro-differential equations with Poisson jump is obtained by establishing an integral inequality.     

Continuous solutions of volterra integral equations with curvilinear stochastic integrals

The existence and uniqueness of a continuous solution of a stochastic integral equation with curvilinear integrals is proved.Translated from Teoriya Sluchainykh Protsessov, No. 16, pp. 38–48, 1988.     

Itô's formula for linear fractional PDEs

In this paper, we introduce a stochastic integral with respect to the solution X of the fractional heat equation on [0,1], interpreted as a divergence operator. This allows to use the techniques of the Malliavin calculus in order to establish an Itô-type formula for the process X.     

On a p-stable multiple integral. I

Summary We construct a multiple p-stable stochastic integral as a Dunford type integral with respect to a L ^q -valued vector measure, 1qp2.     

On a new set-valued stochastic integral with respect to semimartingales and its applications

We present a new approach to a concept of a set-valued stochastic integral with respect to semimartingales. Such an integral, called set-valued stochastic up-trajectory integral, is compatible with the decomposition of the semimartingale. Some properties of this integral are stated. We show applicability of the new integral in set-valued stochastic integral equations driven by multidimensional semimartingales. The uniqueness theorem is presented. Then we extend the notion of the set-valued stochastic up-trajectory integral to definition of a fuzzy stochastic up-trajectory integral with respect to semimartingales. A result on uniqueness of a solution to fuzzy stochastic integral equations incorporating the new fuzzy stochastic up-trajectory integral driven by the multidimensional semimartingale is stated.     

On a p-stable multiple integral. II

Summary We construct a multiple strictly p-stable stochastic integral as a Dunford type integral with respect to a L ^q -valued vector measure, 1qp2.     

Duality and Multiplicative Stochastic Processes on Quantum Groups

An analogue of McKean's stochastic product integral is introduced and used to define stochastic processes with independent increments on quantum groups. The explicit form of the dual pairing (q-analogue of the exponential map) is calculated for a large class of quantum groups. The constructed processes are shown to satisfy generalized Feynman-Kac type formulas, and polynomial solutions of associated evolution equations are introduced in the form of Appell systems. Explicit calculations for Gauss and Poisson processes complete the presentation.