Stability of stochastic integrals under change of filtration

Let (Ω, ,P) be a probability space equipped with two filtrations { _t} and { _t} satisfying the usual conditions. Assume that X is a semimartingale and that h is locally bounded and predictable for each of the two filtrations { _t} and { _t}. New examples of such processes are given. Utilizing and extending partial results of Zheng (1982), this paper extends the available results on the relationship between the stochastic integral processes ∫^th_s dX_s taken respectively in the sense of { _t} and of { _t}. In particular, it is shown that these stochastic integrals differ at most by a continuous process with quadratic variation defined and equal to 0. If both stochastic integrals are { _t∩ _t} semimartingales, then it is proved that the stochastic integral ∫^th_s dX _s taken in { _t} sense is indistinguishable from that taken in { _t} sense.     

Estimate for stochastic integrals

Lithuanian Mathematical Journal -     

Frechet-valued martingales and stochastic integrals

Stochastic processes with values in a separable Frechet space whose a itinuous linear functional are real-valued square integrable martingales are investigated. The coordinate measures on the Fréchet space are obtained from cylinder set measures on a Hilbert space that is dense in the Fréchet space. Real-valued stochastic integrals are defined from the Fréchet-valued martingales using integrands from the topological dual of the aforementioned Hilbert space. An increasing process with values in the self adjoint operators on the Hilbert space plays a fundamental role in the definition of stochastic integrals. For Banach-valued Brownian motion the change of variables formula of K. Itô is generalized. A converse to the construction of the measures on the Fréchet space from cylinder set measures on a Hilbert space is also obtained.     

Complex valued multiparameter stochastic integrals

In this paper we will consider the properties of various stochastic integrals over a complex valued, two parameter Wiener process. Integrals of this type exhibit pleasant features which do not appear within the real framework. They are stable under approximation and viewed in relation with analytic functions, they typically satisfy an ordinary chain rule. This in turn gives rise to several nice representation formulas.     

Invariance principles for stochastic area and related stochastic integrals

Given an antisymmetric kernel K (K(z, z′) = ?K(z′, z)) and i.i.d. random variates Z_n, n?1, such that EK²(Z₁, Z₂)<∞, set A_n = ∑₁?i?j?nK(Z_i,Z_j), n?1. If the Z_n's are two-dimensional and K is the determinant function, A_n is a discrete analogue of Paul Lévy's so-called stochastic area. Using a general functional central limit theorem for stochastic integrals, we obtain limit theorems for the A_n's which mirror the corresponding results for the symmetric kernels that figure in theory of U-statistics.     

Quantum stochastic calculus

An introduction to quantum stochastic calculus in symmetric Fock spaces from the point of view of the theory of stochastic processes. Among the topics discussed are the quantum Itô formula, applications to probability representation of solutions of differential equations, extensions of dynamical semigroups. New algebraic expressions are given for the chronologically ordered exponential functions generated by stochastic semigroups in classical probability theory.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 36, pp. 3–28, 1990.     

Extended Thorin classes and stochastic integrals

Extended Thorin classes T ^ϰ (R ^d ), ϰ > 0, of infinitely divisible probability laws on R ^d are defined and analytically characterized in [6]. Using general results from [8] and [9], in this paper, we derive a stochastic integral representation of these classes. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 4, pp. 497–503, October–December, 2007.     

Laplace approximation for stochastic line integrals

We prove a precision of large deviation principle for current-valued processes such as shown in Bolthausen et al. (Ann Probab 23(1):236–267, 1995) for mean empirical measures. The class of processes we consider is determined by the martingale part of stochastic line integrals of 1-forms on a compact Riemannian manifold. For the pair of the current-valued process and mean empirical measures, we give an asymptotic evaluation of a nonlinear Laplace transform under a nondegeneracy assumption on the Hessian of the exponent at equilibrium states. As a direct consequence, our result implies the Laplace approximation for stochastic line integrals or periodic diffusions. In particular, we recover a result in Bolthausen et al. (Ann Probab 23(1):236–267, 1995) in our framework.     

A generalization of operator stochastic integrals

We generalize the method for construction of operator stochastic integrals suggested by Berezanskii, Zhernakov, and Us. We extend the class of integrable commutative quantum processes and study properties of corresponding integrals.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 425–429, April, 1994.This work was partially supported by the Ukrainian State Committee on Science and Technology.     

Some properties of set-valued stochastic integrals

The present paper is devoted to properties of set-valued stochastic integrals defined as some special type of set-valued random variables. In particular, it is shown that if the probability base is separable or probability measure is nonatomic then defined set-valued stochastic integrals can be represented by a sequence of Itô?s integrals of nonanticipative selectors of integrated set-valued processes. Immediately from Michael?s continuous selection theorem it follows that the indefinite set-valued stochastic integrals possess some continuous selections. The problem of integrably boundedness of set-valued stochastic integrals is considered. Some remarks dealing with stochastic differential inclusions are also given.     

Quantitative approximation of certain stochastic integrals

We approximate certain stochastic integrals, typically appearing in Stochastic Finance, by stochastic integrals over integrands, which are path-wise constant within deterministic, but not necessarily equidistant, time intervals. We ask for rates of convergence if the approximation error is considered in L 2 . In particular, we show that by using non-equidistant time nets, in contrast to equidistant time nets, approximation rates can be improved considerably.     

Generalized Brownian functionals and stochastic integrals

The purpose of this paper is two-fold; i) a new class of generalized Brownian functionals, in fact generalized linear functionals, is introduced and ii) generalized stochastic integrals based on creation operators are discussed. These topics are in line with the causal calculus of Brownian functionals.Communicated by H. H. Kuo