CLARK-OCONE TYPE FORMULAS ON THE SPACES OF NONREGULAR GENERALIZED FUNCTIONS IN THE LÉVY WHITE NOISE ANALYSIS N. A. Kachanovsky Methods of Functional Analysis and Topology, 2025 In the classical Gaussian analysis the Clark-Ocone formula can be written in the form \[ F=\mathbf EF+\int\mathbf E\big(\partial_t F|_{\mathcal F_t}\big)dW_t, \] where a function (a random variable) $F$ is square integrable with respect to the Gaussian measure and differentiable by Hida; $\mathbf E$ denotes the expectation; $\mathbf E\big(\circ|_{\mathcal F_t}\big)$---the conditional expectation with respect to the $\sigma$-algebra $\mathcal F_t$ that is generated by a Wiener process $W$ up to the point of time $t$; $\partial_\cdot F$ is the Hida derivative of $F$; $\int\circ (t)dW_t$ denotes the Itô stochastic integral over a Wiener process. This formula has many applications, in particular, in the stochastic analysis and in the financial mathematics. <br><br> In this paper we generalize the Clark-Ocone formula to the spaces $(\mathcal H_{-\tau})_{-q}$ of nonregular generalized functions in the Lévy white noise analysis. More exactly, we prove that any element of $(\mathcal H_{-\tau})_{-q}$ can be represented as a sum of its expectation and a result of stochastic integration over a Lévy process of some generalized function, and construct Clark-Ocone type formulas on $(\mathcal H_{-\tau})_{-q}$ and on its subsets.
Wick multiplication and its relationship with integration and stochastic differentiation on spaces of nonregular test functions in the Lévy white noise analysis N.A. Kachanovsky Carpathian Mathematical Publications, 2024 We deal with spaces of nonregular test functions in the Lévy white noise analysis, which are constructed using Lytvynov's generalization of a chaotic representation property. Our goal is to study properties of a natural multiplication $-$ a Wick multiplication on these spaces, and to describe the relationship of this multiplication with integration and stochastic differentiation. More exactly, we establish that the Wick product of nonregular test functions is a nonregular test function; show that when employing the Wick multiplication, it is possible to take a time-independent multiplier out of the sign of a generalized stochastic integral; establish an analog of this result for a Pettis integral (a weak integral); obtain a representation of the generalized stochastic integral via formal Pettis integral from the Wick product of the original integrand by a Lévy white noise; and prove that the operator of stochastic differentiation of first order on the spaces of nonregular test functions satisfies the Leibnitz rule with respect to the Wick multiplication.
On Wick calculus and its relationship with stochastic integration on spaces of regular test functions in the Lévy white noise analysis N.A. Kachanovsky Carpathian Mathematical Publications, 2022 We deal with spaces of regular test functions in the Lévy white noise analysis, which are constructed using Lytvynov's generalization of a chaotic representation property. Our aim is to study properties of Wick multiplication and of Wick versions of holomorphic functions, and to describe a relationship between Wick multiplication and integration, on these spaces. More exactly, we establish that a Wick product of regular test functions is a regular test function; under some conditions a Wick version of a holomorphic function with an argument from the space of regular test functions is a regular test function; show that when employing the Wick multiplication, it is possible to take a time-independent multiplier out of the sign of an extended stochastic integral with respect to a Lévy process; establish an analog of this result for a Pettis integral (a weak integral); obtain a representation of the extended stochastic integral via formal Pettis integral from the Wick product of the original integrand by a Lévy white noise. As an example of an application of our results, we consider an integral stochastic equation with Wick multiplication.
Interconnection between wick multiplication and integration on spaces of nonregular generalized functions in the lévy white noise analysis N.A. Kachanovsky, T.O. Kachanovska Carpathian Mathematical Publications, 2019 We deal with spaces of nonregular generalized functions in the Lévy white noise analysis, which are constructed using Lytvynov's generalization of a chaotic representation property. Our aim is to describe a relationship between Wick multiplication and integration on these spaces. More exactly, we show that when employing the Wick multiplication, it is possible to take a time-independent multiplier out of the sign of an extended stochastic integral; establish an analog of this result for a Pettis integral (a weak integral); and prove a theorem about a representation of the extended stochastic integral via the Pettis integral from the Wick product of the original integrand by a Lévy white noise. As examples of an application of our results, we consider some stochastic equations with Wick type nonlinearities.
Some remarks on operators of stochastic differentiation in the Lévy white noise analysis Methods of Functional Analysis and Topology, 2017
An extended stochastic integral and a wick calculus on parametrized Kondratiev-type spaces of meixner white noise N. A. KACHANOVSKY Infinite Dimensional Analysis Quantum Probability and Related Topics, 2008 Using a general approach that covers the cases of Gaussian, Poissonian, Gamma, Pascal and Meixner measures, we consider an extended stochastic integral and construct elements of a Wick calculus on parametrized Kondratiev-type spaces of generalized 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 of studying the properties of the extended (Skorohod) stichastic integral subject to the particular spaces under consideration; and of studying the properties of a Wick product and Wick versions of holomorphic functions on the parametrized Kondratiev-type spaces. These results are necessary, in particular, in order to describe properties of solutions of normally ordered white noise equations in the "Meixner analysis".
Clark-Ocone type formulas on the spaces of nonregular generalized functions in the Lévy white noise analysis NA Kachanovsky Methods of Functional Analysis and Topology 31 (02), 117-135 , 2025 2025
Elements of Lévy Analysis in the Spaces of Irregular Test and Generalized Functions N Kachanovsky Ukrainian Mathematical Journal 76 (12), 1967-2001 , 2025 2025 Citations: 1
Wick multiplication and its relationship with integration and stochastic differentiation on spaces of nonregular test functions in the Lévy white noise analysis NA Kachanovsky Carpathian Mathematical Publications 16 (1), 61-83 , 2024 2024 Citations: 3
ЕЛЕМЕНТИ АНАЛІЗУ ЛЕВІ НА ПРОСТОРАХ НЕРЕГУЛЯРНИХ ОСНОВНИХ ТА УЗАГАЛЬНЕНИХ ФУНКЦІЙ N KACHANOVSKY UKRAINS’KYI MATEMATYCHNYI ZHURNAL Учредители: Institute of Mathematics … , 2024 2024
On Wick calculus and its relationship with stochastic integration on spaces of regular test functions in the Lévy white noise analysis NA Kachanovsky Carpathian Mathematical Publications 14 (1), 194-212 , 2022 2022 Citations: 8
On the relationship between Wick calculus and stochastic integration in the Lévy white noise analysis MM Frei, NA Kachanovsky European Journal of Mathematics 6 (1), 179-196 , 2020 2020 Citations: 6
Interconnection between Wick multiplication and integration on spaces of nonregular generalized functions in the Lévy white noise analysis NA Kachanovsky, T Kachanovska Карпатські математичні публікації, 70-88 , 2019 2019 Citations: 8
On Wick calculus on spaces of nonregular generalized functions of Lévy white noise analysis NA Kachanovsky Карпатські математичні публікації, 114-132 , 2018 2018 Citations: 10
Some remarks on operators of stochastic differentiation in the Lévy white noise analysis MM Frei, NA Kachanovsky Methods of Functional Analysis and Topology 23 (4), 320-345 , 2017 2017 Citations: 11
Operators of stochastic differentiation on spaces of nonregular generalized functions of Lévy white noise analysis NA Kachanovsky Карпатські математичні публікації, 83-106 , 2016 2016 Citations: 14
Operators of stochastic differentiation on spaces of nonregular test functions of Lévy white noise analysis NA Kachanovsky Methods of Functional Analysis and Topology 21 (04), 336-360 , 2015 2015 Citations: 20
On operators of stochastic differentiation on spaces of regular test and generalized functions of L\'evy white noise analysis MM Dyriv, NA Kachanovsky Карпатські математичні публікації, 212-229 , 2014 2014 Citations: 18
On Kondratiev spaces of test functions in the non-Gaussian infinite-dimensional analysis NA Kachanovsky Methods of Functional Analysis and Topology 19 (4), 301-309 , 2013 2013 Citations: 4
Розширені стохастичні інтеграли за процесом Леві на просторах узагальнених функцій М Качановський Математичний вісник Наукового товариства ім. Шевченка 10 (1-2), 169-188 , 2013 2013
Stochastic integrals with respect to a Levy process and stochastic derivatives on spaces of regular test and generalized functions MM Dyriv, NA Kachanovsky Наукові вісті Національного технічного університету України Київський … , 2013 2013 Citations: 17
Extended stochastic integrals with respect to a Lévy process on spaces of generalized functions NA Kachanovsky Математичний вісник Наукового товариства ім. Шевченка, 169-188 , 2013 2013 Citations: 29
On extended stochastic integrals with respect to Lévy processes NA Kachanovsky Карпатські математичні публікації, 256-278 , 2013 2013 Citations: 32
Стохастичні інтеграли по процесу Леві та стохастичні похідні на просторах регулярних основних і узагальнених функцій MM Dyriv, NA Kachanovsky Наукові вісті Національного технічного університету України" Київський … , 2013 2013
On Stochastic Integral, Wick Calculus and Generalized Stochastic Equations NA Kachanovsky Operator Theory and Related Topics: Proceedings of the Mark Krein … , 2012 2012
Stochastic Integral and Stochastic Derivative Connected with a Lévy Progress. NA Kachanovsky Naukovi visti NTUU-KPI 84 (4) , 2012 2012 Citations: 9
MOST CITED SCHOLAR PUBLICATIONS
On an extended stochastic integral and the Wick calculus on the connected with the generalized Meixner measure Kondratiev-type spaces NA Kachanovsky Methods of Functional Analysis and Topology 13 (4), 338-379 , 2007 2007 Citations: 40
On extended stochastic integrals with respect to Lévy processes NA Kachanovsky Карпатські математичні публікації, 256-278 , 2013 2013 Citations: 32
Extended stochastic integrals with respect to a Lévy process on spaces of generalized functions NA Kachanovsky Математичний вісник Наукового товариства ім. Шевченка, 169-188 , 2013 2013 Citations: 29
Stochastic integral of Hitsuda–Skorokhod type on the extended Fock space NA Kachanovsky, VA Tesko Ukrainian Mathematical Journal 61 (6), 873-907 , 2009 2009 Citations: 28
Biorthogonal Appell-like systems in a Hilbert space NA Kachanovsky Methods of Functional Analysis and Topology 2 (3), 36-52 , 1996 1996 Citations: 27
Minimality of Appell-like systems and embeddings of test function spaces in a generalization of white noise analysis NA Kachanovsky, SV Koshkin Methods of Functional Analysis and Topology 5 (3), 13-25 , 1999 1999 Citations: 23
Operators of stochastic differentiation on spaces of nonregular test functions of Lévy white noise analysis NA Kachanovsky Methods of Functional Analysis and Topology 21 (04), 336-360 , 2015 2015 Citations: 20
An extended stochastic integral and a Wick calculus on parametrized Kondratiev-type spaces of Meixner white noise NA Kachanovsky Infinite Dimensional Analysis, Quantum Probability and Related Topics 11 (04 … , 2008 2008 Citations: 19
On operators of stochastic differentiation on spaces of regular test and generalized functions of L\'evy white noise analysis MM Dyriv, NA Kachanovsky Карпатські математичні публікації, 212-229 , 2014 2014 Citations: 18
Stochastic integrals with respect to a Levy process and stochastic derivatives on spaces of regular test and generalized functions MM Dyriv, NA Kachanovsky Наукові вісті Національного технічного університету України Київський … , 2013 2013 Citations: 17
Generalized stochastic derivatives on parametrized spaces of regular generalized functions of Meixner white noise NA Kachanovsky Methods of Functional Analysis and Topology 14 (4), 334-350 , 2008 2008 Citations: 17
Dual Appell-like systems and finite order spaces in non-Gaussian infinite dimensional analysis NA Kachanovsky Methods of Functional Analysis and Topology 4 (02), 41-52 , 1998 1998 Citations: 17
A generalized stochastic derivative on the Kondratiev-type space of regular generalized functions of Gamma white noise NA Kachanovsky Methods of Functional Analysis and Topology 12 (04), 363-383 , 2006 2006 Citations: 16
On the extended stochastic integral connected with the Gamma-measure on an infinite-dimensional space NA Kachanovsky Methods of Functional Analysis and Topology 8 (02), 10-32 , 2002 2002 Citations: 16
On an extended Malliavin derivative connected with the Poisson-and Gamma-measures NA Kachanovsky Methods of Functional Analysis and Topology 9 (3), 213-240 , 2003 2003 Citations: 15
Operators of stochastic differentiation on spaces of nonregular generalized functions of Lévy white noise analysis NA Kachanovsky Карпатські математичні публікації, 83-106 , 2016 2016 Citations: 14
Generalized stochastic derivatives on a space of regular generalized functions of Meixner white noise NA Kachanovsky Methods of Functional Analysis and Topology 14 (1), 32-53 , 2008 2008 Citations: 14
On analog of stochastic integral and Wick calculus in non-Gaussian infinite dimensional analysis NA Kachanovsky Methods of Functional Analysis and Topology 3 (3), 1-12 , 1997 1997 Citations: 12
Some remarks on operators of stochastic differentiation in the Lévy white noise analysis MM Frei, NA Kachanovsky Methods of Functional Analysis and Topology 23 (4), 320-345 , 2017 2017 Citations: 11
On Wick calculus on spaces of nonregular generalized functions of Lévy white noise analysis NA Kachanovsky Карпатські математичні публікації, 114-132 , 2018 2018 Citations: 10
