Індекс рубрикатора НБУВ: В162.12 + В171.505

Рубрики:

Узагальнені функції

Стохастичні диференціальні та інтегральні рівняння

Шифр НБУВ: Ж26161 Пошук видання у каталогах НБУВ 

Введено та вивчено узагальнені стохастичні похідні на пов'язаних з білим шумом Майкснера просторах типу Кондратьєва нерегулярних узагальнених функцій. Властивості цих похідних аналогічні властивостям стохастичних похідних у гауссівському аналізі. Як приклад обчислено узагальнену стохастичну похідну розв'язку певного стохастичного рівняння з нелінійністю типу Віка.

We review some recent results connected with stochastic integrals of Hitsuda - Skorokhod type acting on the extended Fock space and its riggings.

Наведено огляд деяких результатів негауссівського аналізу при біортогональному підході та розглянуто елементи аналізу, пов'язаного з узагальненою мірою Майкснера. Основними об'єктами, що розглянуто, є стохастичні інтеграли, оператори стохастичного диференціювання, елементи віківського числення та споріднені питання.

Індекс рубрикатора НБУВ: В161.501

Шифр НБУВ: Ж16492 Пошук видання у каталогах НБУВ 

We introduce and study a generalized Malliavin derivative on the spaces of functions that are square integrable with respect to the Poisson measure and the Gamma-measure, and on the Kondratiev generalized functions spaces that are connected with these measures.

We introduce a generalized stochastic (Malliavin) derivative on the spaces of functions that are square integrable with respect to the so-called coloured noise measures and on the corresponding Kondratiev-type regular generalized spaces of functions (in the infinite-dimensional case). Then we study the main properties of this derivative.

We introduce and study a generalized stochastic derivative on the Kondratiev-type space of regular generalized functions of Gamma white noise. Properties of this derivative are quite analogous to the properties of the stochastic derivative in the Gaussian analysis. As an example we calculate the generalized stochastic derivative of the solution of some stochastic equation with Wick-type nonlinearity.

We introduce and study generalized stochastic derivatives on a Kondratiev-type space of regular generalized functions of Meixner white noise. Properties of these derivatives are quite analogous to the properties of the stochastic derivatives in the Gaussian analysis. As an example we calculate the generalized stochastic derivative of the solution of some stochastic equation with a Wick-type nonlinearity.

We introduce and study Hida-type stochastic derivatives and stochastic differential operators on the parametrized Kondratiev-type spaces of regular generalized functions of Meixner white noise. In particular, we study the interconnection between the stochastic integration and differentiation. Our researches are based on the general approach that covers the Gaussian, Poissonian, Gamma, Pascal and Meixner cases.

We introduce an extended stochastic integral and construct elements of the Wick calculus on the Kondratiev-type spaces of regular and nonregular generalized functions, study the interconnection between the extended stochastic integration and the Wick calculus, and consider examples of stochastic equations with Wick-type nonlinearity. Our researches are based on the general approach that covers the Gaussian, Poissonian, Gamma, Pascal and Meixner analyses.

Using a general approach that covers the cases of Gaussian, Poissonian, Gamma, Pascal and Meixner measures, we construct elements of a Wick calculus on parametrized Kondratiev-type spaces of test functions; consider the interconnection between the extended stochastic integration and the Wick calculus; and give an example of a stochastic equation with a Wick-type nonlinearity. The main results consist in studying properties of a Wick product and Wick versions of holomorphic functions on the parametrized Kondratiev-type spaces of test functions. These results are necessary, in particular, in order to describe properties of solutions of stochastic equations with Wick type nonlinearities in the "Meixner analysis".

Using a general approach that covers the cases of Gaussian, Poissonian, Gamma, Pascal and Meixner measures on an infinite-dimensional space, we construct a general integration by parts formula for analysis connected with each of these measures. Our consideration is based on the constructions of the extended stochastic integral and the stochastic derivative that are connected with the structure of the extended Fock space.

Індекс рубрикатора НБУВ: В171.521

Рубрики:

Гаусові випадкові процеси

Шифр НБУВ: Ж41243 Пошук видання у каталогах НБУВ 

Індекс рубрикатора НБУВ: В161.619 + В162.1

Рубрики:

Диференціальні рівняння нескінченного порядку і нескінченні системи диференційних рівнянь

Лінійні, лінійні нормовані і лінійні топологічні простори

Шифр НБУВ: Ж41243 Пошук видання у каталогах НБУВ 

The operators of stochastic differentiation, which are closely related with the extended Skorohod stochastic integral and with the Hida stochastic derivative, play an important role in the classical (Gaussian) white noise analysis. In particular, these operators can be used in order to study properties of the extended stochastic integral and of solutions of stochastic equations with Wick-type nonlinearities. During recent years the operators of stochastic differentiation were introduced and studied, in particular, in the framework of the Meixner white noise analysis, and on spaces of regular test and generalized functions of the Levy white noise analysis. In this paper we make the next step: introduce and study operators of stochastic differentiation on spaces of test functions that belong to the so-called nonregular rigging of the space of square integrable with respect to the measure of a Levy white noise functions, using Lytvynov's generalization of the chaotic representation property. This can be considered as a contribution in a further development of the Levy white noise analysis.

Operators of stochastic differentiation, which are closely related with the extended Skorohod stochastic integral and with the Hida stochastic derivative, play an important role in the Gaussian white noise analysis. In particular, these operators can be used in order to study some properties of the extended stochastic integral and of solutions of so-called normally ordered stochastic equations. During recent years, operators of stochastic differentiation were introduced and studied, in particular, on spaces of regular and nonregular test and generalized functions of the Levy white noise analysis, in terms of Lytvynov's generalization of the chaotic representation property. But, strictly speaking, the existing theory in the "regular case" is incomplete without one more class of operators of stochastic differentiation, in particular, the mentioned operators are required in calculation of the commutator between the extended stochastic integral and the operator of stochastic differentiation. In the present paper we introduce this class of operators and study their properties. In addition, we establish a relation between the introduced operators and the corresponding operators on the spaces of nonregular test functions. The researches of the paper can be considered as a contribution to a further development of the Levy wliite noise analysis.