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We present an approach to proving parabolic Aleksandrov estimates with mixed norms for stochastic integrals with singular "moderated" drift. [ABSTRACT FROM AUTHOR]

We prove that the set of laws of stochastic integrals H ·W, where W is a multidimensional Wiener process and H2 takes values in a compact convex subset D of the set of symmetric positive semidefinite matrices, is weakly dense in the set of laws of martingales M with d⟨M⟩/dt taking values in D. [ABSTRACT FROM AUTHOR]

In this paper we propose a method for finding an integrand in the Clark stochastic integral representation of quadratic integrable Brownian functional F with Malliavin differentiable conditional mathematical expectation E[F|JtB], t < T, where (JtB) is a natural Brownian filtration. The proposed method allows us to determine the explicit form of the integrand in the case when F has no Malliavin derivative. We give several illustrations as examples. [ABSTRACT FROM AUTHOR]

We derive exponential upper bounds for the distribution tails of multiple stochastic integrals of bounded nonrandom functions. In this paper we consider integrating processes from certain wide classes. Some important examples of integrating processes are given. Multiple stochastic integrals with respect to increments of such processes arise in statistical applications. [ABSTRACT FROM AUTHOR]

A minimum contrast criterion for consistently estimating the spatial fractal dimension in the wavelet domain is presented in [M. D. Ruiz-Medina and R. M. Crujeiras, Comm. Statist. Theory Methods., 40 (2011), pp. 3599–3613] for a class of semiparametric fractal Gaussian random fields, which includes fractional Brownian motion and Riesz–Bessel motion as particular cases. In this work, asymptotic normality of this minimum contrast estimator is obtained using central limit results for multiple stochastic integrals (see [N. Nualart and G. Peccati, Ann. Probab., 33 (2005), pp. 91–115]). [ABSTRACT FROM AUTHOR]

A generalized stochastic integral of predictable generalized random processes (predictable random processes) with respect to a real Wiener process is defined. The question of the existence of the stochastic integral in a Banach space is reduced to the problem of decomposability of the generalized random element. The stochastic differential equation for generalized random processes is introduced and existence and uniqueness of solutions are developed. As a consequence the corresponding results on stochastic differential equations in a Banach space are given. [ABSTRACT FROM AUTHOR]

This paper contains sufficient conditions of measurability with respect to a parameter of the limit of a sequence of random fields with trajectories in D. Measurability with respect to a parameter of the quadratic variation of the two-parameter strong martingale was proved. We obtain sufficient conditions of measurability with respect to a parameter of the stochastic integral with respect to the two-parameter strong martingale. [ABSTRACT FROM AUTHOR]

By using multiple Wiener-Itô stochastic integrals, we study the cubic variation of a class of self-similar stochastic processes with stationary increments (the Rosenblatt process with self-similarity order H∊(½, 1)). This study is motivated by statistical purposes. We prove that this renormalized cubic variation satisfies a noncentral limit theorem, and its limit is (in the L²(Ω) sense) still the Rosenblatt process. [ABSTRACT FROM AUTHOR]

Symmetric integrals of the Stieltjes type for an arbitrary continuous function and which are determinate versions of the Stratonovich stochastic integrals are defined. It is shown that the solution of a pathwise analogue of stochastic differential equations with a symmetric integral and Itô stochastic differential equations is reduced to a solution of a system of ordinary differential equations. The relation between improper symmetric integrals, which is extended to symmetric integrals and Hellinger integrals, is established. [ABSTRACT FROM AUTHOR]

In the first part of this paper [A. N. Shiryaev and M. Yor, Theory Probab. Appl., 48 (2004), pp. 304–313], a method of obtaining stochastic integral representations of functionals S(w) of Brownian motion B = (Bt)t≧0 was stated. Functionals maxt≦T Bt and maxt≦T-a Bt, where T-a = inf{t: Bt = -a}, a > 0, were considered as an illustration. In the present paper we state another derivation of representations for these functionals and two proofs of representation for functional maxt≦gT St, where (non-Markov time) gT = sup{0 ≦ t ≦ T: Bt = 0} are given. [ABSTRACT FROM AUTHOR]

In the paper the limiting behavior of finite sums with averaging containing Lévy processes is considered. For this purpose a new class of stochastic integrals for Lévy processes including, in particular, the Itô integral, Stratonovich integral, and others, is defined. By using these integrals complete classification of the limiting behavior of the considered sums is given. [ABSTRACT FROM AUTHOR]

In this paper the construction of a stochastic integral of a nonrandom function is suggested without the classical orthogonality condition of the noise. This construction includes some known constructions of univariate and multiple stochastic integrals. Conditions providing the existence of this integral are specified for noises generated by random processes with nonorthogonal increments from certain classes which are rich enough. [ABSTRACT FROM AUTHOR]

This paper introduces the concept of σ-localization, which is a generalization of localization in the general theory of stochastic processes. The σ-localized class derived from the set of martingales is the class of σ-martingales, which plays an important role in mathematical finance. These processes and the corresponding σ-martingale measures are considered in detail. By extending the stochastic integral with respect to compensated random measures, a canonical representation of σ-martingales as for local martingales is derived. [ABSTRACT FROM AUTHOR]

The definition of a generalized additive homogeneous functional of a Wiener process is introduced. It is shown that a generalized functional is uniquely specified by its characteristics. In this case the functions from the Schwartz space S* of slowly growing generalized functions play the role of generating functions. [ABSTRACT FROM AUTHOR]

By the Dambis-Dubins-Schwarz theorem, any stochastic integral M ≔&inte;[SUP.],[SSUB0]H[SUBs]Dw[sSUBs] of Brownian motion can be written as a time-changed Brownian motion, i.e., M = (Ŵ[SUB&Tcirc;[SUBt]])[SUBt∊[SUBR+]] for some Brownian motion (Ŵ[SUBθ])[SUBθ∊R[SUB+]] and some time change (&Tcirc;[SUBt])[SUBt∊R[SUB+]. In [J. Jacod and A. Shiryaev, Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin-Heidelberg, 1987] and [O. Kallenberg, Stochastic Process. Appl., 40 (1992), pp. 199-223] it is shown that in this statement Brownian motion can be replaced with (symmetric) α-stable Léevy motion. Using the cumulant process of a semimartingale, we give new short proofs. Moreover, we show that the statement cannot be extended to any other Léevy processes. [ABSTRACT FROM AUTHOR]

The theme of providing predictable criteria for absolute continuity and for mutual singularity of two density processes on a filtered probability space is extensively studied, e.g., in the monograph by J. Jacod and A. N. Shiryaev [Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin, 1987]. While the issue of absolute continuity is settled there in full generality, for the issue of mutual singularity one technical difficulty remained open [J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin, 1987, p. 210]: "We do not know whether it is possible to derive a predictable criterion (necessary and sufficient condition) for P'[SUBT] ⊥ P[SUBT],...". It turns out that there are two answers to this question raised in the monograph of J. Jacod and A. N. Shiryaev: On the negative side, we give an easy example showing that in general the answer is no, even when we use a rather wide interpretation of the concept of "predictable criterion." The difficulty comes from the fact that the density process of a probability measure P with respect to another measure P' may suddenly jump to zero. On the positive side, we can characterize the set where P' becomes singular with respect to P -- provided this happens in a continuous way rather than suddenly -- as the set where the Hellinger process diverges, which certainly is a "predictable criterion." This theorem extends results in the monograph of J. Jacod and A. N. Shiryaev. [ABSTRACT FROM AUTHOR]

A condition is given in terms of the absolute moments of stochastic integrals for two stochastic processes, continuous in probability with independent stationary symmetric increments, to be identical. characterization, stochastic integral, stochastic process, absolute moment [ABSTRACT FROM AUTHOR]

Information on the Fifth International Conference titled Limit Theorems in Probability Theory and Their Applications, which was held at the Sobolev Institute of Mathematics in Novosibirsk, Russia from August 15-18, 2011 is presented. Topics include central limit theorem, random walks, and stochastic integrals. The conference featured several speakers including mathematicians A. A. Borovkov and T. Mikosh, and author I. A. Ibragimov.