# N. A. Kachanovsky

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### Some remarks on operators of stochastic differentiation in the Lévy white noise analysis

MFAT **23** (2017), no. 4, 320-345

320-345

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 Lévy 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 Lévy white noise analysis.

### Operators of stochastic differentiation on spaces of nonregular test functions of Lévy white noise analysis

MFAT **21** (2015), no. 4, 336-360

336-360

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.

### On Kondratiev spaces of test functions in the non-Gaussian infinite-dimensional analysis

MFAT **19** (2013), no. 4, 301-309

301-309

A blanket version of the non-Gaussian analysis under the so-called bior hogo al approach uses the Kondratiev spaces of test functions with orthogonal bases given by a generating function $Q\times H \ni (x,\lambda)\mapsto h(x;\lambda)\in\mathbb C$, where $Q$ is a metric space, $H$ is some complex Hilbert space, $h$ satisfies certain assumptions (in particular, $h(\cdot;\lambda)$ is a continuous function, $h(x;\cdot)$ is a holomorphic at zero function). In this paper we consider the construction of the Kondratiev spaces of test functions with orthogonal bases given by a generating function $\gamma(\lambda)h(x;\alpha(\lambda))$, where $\gamma :H\to\mathbb C$ and $\alpha :H\to H$ are holomorphic at zero functions, and study some properties of these spaces. The results of the paper give a possibility to extend an area of possible applications of the above mentioned theory.

### Clark-Ocone type formulas on spaces of test and generalized functions of Meixner white noise analysis

MFAT **18** (2012), no. 2, 160-175

160-175

In the classical Gaussian analysis the Clark-Ocone formula can be written in the form $$ F=\mathbf EF+\int\mathbf E_t\partial_t FdW_t, $$ where the function (the random variable) $F$ is square integrable with respect to the Gaussian measure and differentiable by Hida; $\mathbf E$ denotes the expectation; $\mathbf E_t$ denotes the conditional expectation with respect to the full $\sigma$-algebra 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 It\^o stochastic integral with respect to the Wiener process. This formula has applications in the stochastic analysis and in the financial mathematics. In this paper we generalize the Clark-Ocone formula to spaces of test and generalized functions of the so--called Meixner white noise analysis, in which instead of the Gaussian measure one uses the so--called generalized Meixner measure $\mu$ (depending on parameters, $\mu$ can be the Gaussian, Poissonian, Gamma measure etc.). In particular, we study properties of integrands in our (Clark-Ocone type) formulas.

### Notes on Wick calculus on parametrized test functions spaces of Meixner white noise

MFAT **17** (2011), no. 2, 150-167

150-167

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".

### The integration by parts formula in the Meixner white noise analysis

MFAT **16** (2010), no. 1, 6-16

6-16

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.

### Generalized stochastic derivatives on parametrized spaces of regular generalized functions of Meixner white noise

MFAT **14** (2008), no. 4, 334-350

334-350

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.

### Generalized stochastic derivatives on a space of regular generalized functions of Meixner white noise

MFAT **14** (2008), no. 1, 32-53

32-53

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.

### On an extended stochastic integral and the Wick calculus on the connected with the generalized Meixner measure Kondratiev-type spaces

MFAT **13** (2007), no. 4, 338-379

338-379

We introduce an extended stochastic integral and construct elements of the Wick calculus on the Kondratiev-type spaces of regular and nonregular gene alized 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.

### A generalized stochastic derivative on the Kondratiev-type space of regular generalized functions of Gamma white noise

MFAT **12** (2006), no. 4, 363-383

363-383

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.

### A generalized stochastic derivative connected with coloured noise measures

MFAT **10** (2004), no. 4, 11-29

11-29

### On an extended Malliavin derivative connected with the Poisson- and Gamma-measures

MFAT **9** (2003), no. 3, 213-240

213-240

### On the extended stochastic integral connected with the gamma-measure on an infinite-dimensional space

MFAT **8** (2002), no. 2, 10-32

10-32

### On biorthogonal approach to construction of non-Gaussian analysis and application to Poisson analysis on configuration space

MFAT **6** (2000), no. 2, 13-21

13-21

### Minimality of Appell-like systems and embeddings of test function spaces in a generalization of white noise analysis

N. A. Kachanovsky, S. V. Koshkin

MFAT **5** (1999), no. 3, 13-25

13-25

### Wick algebras and quantum free fields in Gaussian and non-Gaussian analysis

MFAT **4** (1998), no. 4, 22-39

22-39

### Dual Appell-like systems and finite order spaces in non-Gaussian infinite dimensional analysis

MFAT **4** (1998), no. 2, 41-52

41-52

### On analog of stochastic integral and Wick calculus in non-Gaussian infinite dimensional analysis

MFAT **3** (1997), no. 3, 1-12

1-12

### Biorthogonal Appell-like systems in a Hilbert space

MFAT **2** (1996), no. 3, 36-52

36-52