Stochastic integral of Hitsuda–Skorokhod type on the extended Fock space

Kachanovsky, N. A.; Tesko, Volodymyr

doi:10.1007/s11253-009-0257-2

Cited by 7 publications

(8 citation statements)

References 38 publications

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“…can be obtained by direct calculation with use (15), (6), (9), (19) and the orthogonality of polynomials (14) in L 2 (R, ν), see also Lemma 4.1 in [22].…”

Section: Constructions and Some Properties Of Extended Stochastic Intmentioning

confidence: 99%

“…Since integrals (32) and (42) coincide with the extended stochastic integral (26), these integrals are extensions of the Itô stochastic integral. Note that for integral (42) this fact can be proved by analogy with the corresponding proof in the "Meixner analysis" [17] (see also [19]) with using results of [4].…”

Section: Lemma For Each Fmentioning

confidence: 99%

“…(here we used (19)), hence f (n) generates an equivalence class f (n) ∈ H (n+1) ext and estimate (38) is fulfilled. Letġ…”

mentioning

confidence: 99%

“…Without loss of generality we can assume thatġ (n) is a symmetric function, therefore one can separate the last argument. Using (18) and (19) one can write…”

mentioning

confidence: 99%

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On extended stochastic integrals with respect to Lévy processes

Kachanovsky

2013

Carpathian Math. Publ.

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Let L be a Lévy process on [0, +∞). In particular cases, when L is a Wiener or Poisson process, any square integrable random variable can be decomposed in a series of repeated stochastic integrals from nonrandom functions with respect to L. This property of L, known as the chaotic representation property (CRP), plays a very important role in the stochastic analysis. Unfortunately, for a general Lévy process the CRP does not hold.There are different generalizations of the CRP for Lévy processes. In particular, under the Itô's approach one decomposes a Lévy process L in the sum of a Gaussian process and a stochastic integral with respect to a Poisson random measure, and then uses the CRP for both terms in order to obtain a generalized CRP for L. The Nualart-Schoutens's approach consists in decomposition of a square integrable random variable in a series of repeated stochastic integrals from nonrandom functions with respect to so-called orthogonalized centered power jump processes, these processes are constructed with using of a cádlág version of L. The Lytvynov's approach is based on orthogonalization of continuous polynomials in the space of square integrable random variables.In this paper we construct the extended stochastic integral with respect to a Lévy process and the Hida stochastic derivative in terms of the Lytvynov's generalization of the CRP; establish some properties of these operators; and, what is most important, show that the extended stochastic integrals, constructed with use of the above-mentioned generalizations of the CRP, coincide.

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“…can be obtained by direct calculation with use (15), (6), (9), (19) and the orthogonality of polynomials (14) in L 2 (R, ν), see also Lemma 4.1 in [22].…”

Section: Constructions and Some Properties Of Extended Stochastic Intmentioning

confidence: 99%

Section: Lemma For Each Fmentioning

confidence: 99%

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On extended stochastic integrals with respect to Lévy processes

Kachanovsky

2013

Carpathian Math. Publ.

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“…Note that some results of the paper can be transferred to weighted symmetric Fock spaces, by analogy with [32]. This gives an opportunity to extend an area of possible applications of these results.…”

Section: Introductionmentioning

confidence: 92%

On Wick calculus on spaces of nonregular generalized functions of Levy white noise analysis

Kachanovsky

2018

Carpathian Math. Publ.

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Development of a theory of test and generalized functions depending on infinitely many variables is an important and actual problem, which is stipulated by requirements of physics and mathematics. One of successful approaches to building of such a theory consists in introduction of spaces of the above-mentioned functions in such a way that the dual pairing between test and generalized functions is generated by integration with respect to some probability measure. First it was the Gaussian measure, then it were realized numerous generalizations. In particular, important results can be obtained if one uses the Lévy white noise measure, the corresponding theory is called the Lévy white noise analysis.In the Gaussian case one can construct spaces of test and generalized functions and introduce some important operators (e.g., stochastic integrals and derivatives) on these spaces by means of a so-called chaotic representation property (CRP): roughly speaking, any square integrable random variable can be decomposed in a series of repeated Itô's stochastic integrals from nonrandom functions. In the Lévy analysis there is no the CRP, but there are different generalizations of this property.In this paper we deal with one of the most useful and challenging generalizations of the CRP in the Lévy analysis, which is proposed by E. W. Lytvynov, and with corresponding spaces of nonregular generalized functions. The goal of the paper is to introduce a natural product (a Wick product) on these spaces, and to study some related topics. Main results are theorems about properties of the Wick product and of Wick versions of holomorphic functions. In particular, we prove that an operator of stochastic differentiation satisfies the Leibniz rule with respect to the Wick multiplication. In addition we show that the Wick products and the Wick versions of holomorphic functions, defined on the spaces of regular and nonregular generalized functions, constructed by means of Lytvynov's generalization of the CRP, coincide on intersections of these spaces.Our research is a contribution in a further development of the Lévy white noise analysis.

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