Ito Stochastic Integrals Research Articles

Quasi-integrals and stochastic integration along sample paths

We construct deterministic analogues of the Stratonovich and Ito stochastic integrals, which coincide with the classical stochastic integrals in the case of the Wiener process.

An Approximation of the Ito and Stratonovich Stochastic Integrals by Elements of a Direct Product of Algebras of Generalized Random Processes

The paper investigates a problem of approximation of multidimensional Ito and Stratonovich stochastic integrals by finite sums of elements of a direct product of algebras of generalized random processes.

Continuity of the Ito stochastic integral in Hilbert spaces

There exists a satisfactory limit theory for stochastic integrals driven by finite dimensional semimartingales. The paper is an attempt to construct a corresponding theory in the case of processes with values in infinite dimensional Hilbert spaces. Limit theorems are given for scalar-and tensor-product integrals and for Hilbert-Schmidt integrands. As a corollary, a criterion for convergence or the tensor quadratic variation is established.

White noise approach to multiparameter stochastic integration

In this paper we will set up the Hida theory of generalized Wiener functionals using S ∗( R d ), the space of tempered distributions on R d , and apply the theory to multiparameter stochastic integration. With the partial ordering on R + d : ( s 1, …, s d ) < ( t 1, …, t d ) if s i < t i , 1 ≤ i ≤ d, the Wiener process W((t 1 …, t d), x) = 〈x, 1 [0,t 1)x … x[0,t d) 〉, ξ ϵ I ∗(T d) is a generalization of a Brownian motion and there is the Wiener-Ito decomposition: L 2( S ∗( R d )) = Σ n = 0 ∞⊕ K n , where K n is the space of n-tuple Wiener integrals. As in the one-dimensional case, there are the continuous inclusions (L 2) + ⊂ L 2(I ∗(R d)) ⊂ (L 2− and ( L 2) − is considered the space of generalized Wiener functionals. We prove that the multidimensional Ito stochastic integral is a special case of an element of ( L 2) −. For d = 2 the Ito integral is not sufficient for representing elements of L 2( S ∗( R 2)). We show that the other stochastic integral involved can also be realized in the Hida setting. For F∈ S ∗( R ) we will define F( W( s, t), x) as an element of ( L 2) − and obtain a generalized Ito formula.

Ito stochastic integral in the dual of a nuclear space

Ito's definition of the stochastic integral with respect to a Wiener process in the dual of a nuclear space is simplified and slightly generalized. This definition yields a completely intrinsic description of the class of random, operator-valued integrands. For a large class of spaces (e.g. for Schwartz distribution spaces) any time-inhomogeneous Wiener process is proved to have a representation as the stochastic integral with respect to a homogeneous (standard) Wiener process. A relation between this definition of stochastic integral and the notion of isometric integral in Hilbert spaces, defined by Metivier, is established.

On Ito Stochastic Integration with Respect to $p$-Stable Motion: Inner Clock, Integrability of Sample Paths, Double and Multiple Integrals

The paper studies in detail the sample paths of Ito-type stochastic integrals with respect to $p$-stable motion $M(t), t \geq 0$. These results, in turn, permit an analysis of the concept of multiple $p$-stable integrals of the form $\int \cdots \int f(t_1,\cdots, t_n)dM(t_1) \cdots dM(t_n)$, and, in particular, a full description of functions of two variables $f(t_1, t_2)$ for which the double stochastic integral $\int \int f(t_1, t_2) dM(t_1) dM(t_2)$ exists.

Diffusion approximations to linear stochastic difference equations with stationary coefficients

It is shown that solutions to linear first-order stochastic difference equations with stationary autocorrelated coefficients converge weakly inD[0,1] to an Ito stochastic integral plus a correction term when the time scale is shifted so that the means, variances, and covariances of the coefficients all approach zero at the same rate. Other limit theorems applicable to different time scale shifts are also given. These results yield two different continuous time limits to a recent model of Roughgarden (1975) for population growth in stationary random environments. One limit, an Ornstein-Uhlenbeck process, is applicable in the presence of rapidly fluctuating autocorrelated environments; the other limit, which is not a diffusion process, applies to the case of slowly varying, highly autocorrelated environments. Other applications in population biology and genetics are discussed.

Diffusion approximations to linear stochastic difference equations with stationary coefficients

It is shown that solutions to linear first-order stochastic difference equations with stationary autocorrelated coefficients converge weakly in D[0,1] to an Ito stochastic integral plus a correction term when the time scale is shifted so that the means, variances, and covariances of the coefficients all approach zero at the same rate. Other limit theorems applicable to different time scale shifts are also given. These results yield two different continuous time limits to a recent model of Roughgarden (1975) for population growth in stationary random environments. One limit, an Ornstein-Uhlenbeck process, is applicable in the presence of rapidly fluctuating autocorrelated environments; the other limit, which is not a diffusion process, applies to the case of slowly varying, highly autocorrelated environments. Other applications in population biology and genetics are discussed.

On the existence of a random solution to a nonlinear perturbed stochastic integral equation

There are two basic classes of random or stochastic integral equations cur ren t ly under s tudy by probabilists and mathemat ical statisticians. Those integral equations involving Ito or Ito-Doob type stochastic integrals and those which can be considered as probabilistic analogues of classical determinist ic integral equations whose formulation involves only the Lebesque integral. I t has been shown [1]-[4], [6]-[11], [13], [14], [16], [17], among others tha t the theory of random integral equations of the l a t t e r ca tegory are ex t remely important in stochastically character izing many physical situations in life sciences and engineering. In this paper we shall be concerned with a stochastic integral equation of the Ito-Doob type. We shall be concerned with a nonlinear stochastic integral equation of the form given by

Asymptotic of the Green's Function of a Riemannian Manifold and Ito's Stochastic Integrals

Quantitative estimates are obtained by comparison with ordinary differential equations associated to a subharmonic exhaustion function q. We associate with q a ratio a, which can be considered as the heat flow in an intrinsic time, and the sup and the inf of a, namely a(+) and a(-), on the level hypersurfaces of q. Then a(+) and a(-) define heat flows on the real line. Comparison between the heat flow on the manifold and heat flows on the line are obtained by stochastic integrals.

Pairing stochastic processes

This paper studies stochastic distributed parameter systems with stochastic inputs. It defines the pairing of two stochastic processes; the Ito stochastic integral is a particular kind of pairing.

The Representation of Functionals of Brownian Motion by Stochastic Integrals

It is known that any functional of Brownian motion with finite second moment can be expressed as the sum of a constant and an Ito stochastic integral. It is also known that homogeneous additive functionals of Brownian motion with finite expectations have a similar representation. This paper extends these results in several ways. It is shown that any finite functional of Brownian motion can be represented as a stochastic integral. This representation is not unique, but if the functional has a finite expectation it does have a unique representation as a constant plus a stochastic integral in which the process of indefinite integrals is a martingale. A corollary of this result is that any martingale (on a closed interval) that is measurable with respect to the increasing family of $\sigma$-fields generated by a Brownian motion is equal to a constant plus an indefinite stochastic integral. Sufficiently well-behaved Frechet-differentiable functionals have an explicit representation as a stochastic integral in which the integrand has the form of conditional expectations of the differential.

Some moment inequalities for stochastic integrals and for solutions of stochastic differential equations

Some inequalities concerning the Ito stochastic integral and solutions of stochastic different equations are obtained.