scholarly journals Wiener-Itô Chaos Expansion of Hilbert Space Valued Random Variables

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
M. A. Alshanskiy
Keyword(s):  
Hilbert Space ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Wiener Process ◽  
Calculus Of Variations ◽  
Hilbert Spaces ◽  
Malliavin Calculus ◽  
Random Variables ◽  
Random Variable ◽  
Chaos Expansion

The notion of n-fold iterated Itô integral with respect to a cylindrical Hilbert space valued Wiener process is introduced and the Wiener-Itô chaos expansion is obtained for a square Bochner integrable Hilbert space valued random variable. The expansion can serve a basis for developing the Hilbert space valued analog of Malliavin calculus of variations which can then be applied to the study of stochastic differential equations in Hilbert spaces and their solutions.

THE GROSS DERIVATIVE AND GENERALIZED RANDOM VARIABLES

1999 ◽  
Vol 02 (03) ◽  
pp. 381-396 ◽  
Author(s):  
FRED ESPEN BENTH
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Random Variables ◽  
Random Variable ◽  
Representation Formula ◽  
Gaussian Random Variable ◽  
Wick Product ◽  
Chaos Expansion ◽  
Schwartz Distributions

We extend the Gross derivative to a space of generalized random variables which have a (formal) chaos expansion with kernels from the space of tempered Schwartz distributions. The extended derivative, which we call the Hida derivative, has to be interpreted in the sense of distributions. Many of the properties of the Gross derivative are proved to hold for the extension as well. In addition, we derive a representation formula for the Hida derivative involving the Wick product and a centered Gaussian random variable. We apply our results to calculate the Hida derivative of a class of stochastic differential equations of Wick type.

Stochastic differential equations

1955 ◽  
Vol 51 (4) ◽  
pp. 663-677 ◽  
Author(s):  
D. A. Edwards ◽  
J. E. Moyal
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Differential Equations ◽  
Strong Convergence ◽  
Stochastic Differential Equations ◽  
Random Variables ◽  
Finite Variance ◽  
Mean Square ◽  
General Banach Space ◽  
Space Approach

The work of which this paper is an account began as a study of differential equations for functions whose values are random variables of finite variance. It was intended that all questions of convergence should be treated from the standpoint of strong convergence in Hilbert space—familiar to probabilists from the writings of Karhunen(11) and Loève(13) as mean-square convergence. The more general Banach-space approach now adopted was made possible by the discovery of a theorem (Theorem 1 of this paper) which Mr D. G. Kendall, its apparent author, kindly communicated to us.

DISSIPATIVE BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS IN INFINITE DIMENSIONS

2006 ◽  
Vol 09 (01) ◽  
pp. 155-168 ◽  
Author(s):  
FULVIA CONFORTOLA
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Hilbert Spaces ◽  
Stochastic Partial Differential Equations ◽  
Existence And Uniqueness ◽  
Backward Stochastic Differential Equations ◽  
Uniqueness Result ◽  
Infinite Dimensions ◽  
Partial Differential

We prove an existence and uniqueness result for a class of backward stochastic differential equations (BSDE) with dissipative drift in Hilbert spaces. We also give examples of stochastic partial differential equations which can be solved with our result.

Existence of Solutions of Abstract Differential Equations in a Local Space

1973 ◽  
Vol 16 (2) ◽  
pp. 239-244
Author(s):  
M. A. Malik
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Differential Equations ◽  
Complex Plane ◽  
Hilbert Spaces ◽  
Scalar Product ◽  
Existence Of Solutions ◽  
Closed Linear Operator ◽  
Abstract Differential Equations ◽  
Local Space

Let H be a Hilbert space; ( , ) and | | represent the scalar product and the norm respectively in H. Let A be a closed linear operator with domain DA dense in H and A* be its adjoint with domain DA*. DA and DA*are also Hilbert spaces under their respective graph scalar product. R(λ; A*) denotes the resolvent of A*; complex plane. We write L = D — A, L* = D — A*; D = (l/i)(d/dt).

Existence of almost periodic solutions for fractional impulsive neutral stochastic differential equations with infinite delay

2019 ◽  
Vol 20 (01) ◽  
pp. 2050003
Author(s):  
Xiao Ma ◽  
Xiao-Bao Shu ◽  
Jianzhong Mao
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Calculus ◽  
Stochastic Differential Equations ◽  
Periodic Solutions ◽  
Infinite Delay ◽  
Almost Periodic Solutions ◽  
Almost Periodic

In this paper, we investigate the existence of almost periodic solutions for fractional impulsive neutral stochastic differential equations with infinite delay in Hilbert space. The main conclusion is obtained by using fractional calculus, operator semigroup and fixed point theorem. In the end, we give an example to illustrate our main results.

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