Chaotic Representation Property Research Articles

Wick multiplication and its relationship with integration and stochastic differentiation on spaces of nonregular test functions in the Lévy white noise analysis

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.

Generalized weighted number operators on functionals of discrete-time normal martingales

Let M be a discrete-time normal martingale that has the chaotic representation property. Then, from the space of square integrable functionals of M, one can construct generalized functionals of M. In this paper, by using a type of weights, we introduce a class of continuous linear operators acting on generalized functionals of M, which we call generalized weighted number (GWN) operators. We prove that GWN operators can be represented in terms of generalized annihilation and creation operators (acting on generalized functionals of M). We also examine commutation relations between a GWN operator and a generalized annihilation (or creation) operator, and obtain several formulas expressing such commutation relations.

On Wick calculus and its relationship with stochastic integration on spaces of regular test functions in the Lévy white noise analysis

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

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.

On the relationship between Wick calculus and stochastic integration in the Lévy white noise analysis

We deal with Lytvynov’s generalization of the chaotic representation property and the corresponding spaces of regular generalized functions in the Levy white noise analysis. Our goal is to find a relationship between the Wick calculus and stochastic integration on these spaces. In particular, we consider the Wick multiplication (a natural multiplication on spaces of generalized functions) under the sign of the stochastic integral, and construct a formal representation of the extended stochastic integral via the Pettis integral, using the Wick product. As application of our results, we consider some stochastic equations with Wick type nonlinearities.

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

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 Levy white noise measure, the corresponding theory is called the Levy 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 Itos stochastic integrals from nonrandom functions. In the Levy 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 Levy 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 Levy white noise analysis.

Wick calculus on spaces of regular generalized functions of Levy white noise analysis

Many objects of the Gaussian white noise analysis (spaces of test and generalized functions, stochastic integrals and derivatives, etc.) can be constructed and studied in terms of so-called chaotic decompositions, based on a chaotic representation property (CRP): roughly speaking, any square integrable with respect to the Gaussian measure random variable can be decomposed in a series of Ito's stochastic integrals from nonrandom functions. In the Levy analysis there is no the CRP (except the Gaussian and Poissonian particular cases). Nevertheless, there are different generalizations of this property. Using these generalizations, one can construct different spaces of test and generalized functions. And in any case it is necessary to introduce a natural product on spaces of generalized functions, and to study related topics. This product is called a Wick product, as in the Gaussian analysis.
 The construction of the Wick product in the Levy analysis depends, in particular, on the selected generalization of the CRP. In this paper we deal with Lytvynov's generalization of the CRP and with the corresponding spaces of regular generalized functions. The goal of the paper is to introduce and to study the Wick product on these spaces, and to consider some related topics (Wick versions of holomorphic functions, interconnection of the Wick calculus with operators of stochastic differentiation). Main results of the paper consist in study of properties of the Wick product and of the Wick versions of holomorphic functions. In particular, we proved that an operator of stochastic differentiation is a differentiation (satisfies the Leibniz rule) with respect to the Wick multiplication.

Clark-Ocone Formula for Generalized Functionals of Discrete-Time Normal Noises

The Clark-Ocone formula in the theory of discrete-time chaotic calculus holds only for square integrable functionals of discrete-time normal noises. In this paper, we aim at extending this formula to generalized functionals of discrete-time normal noises. Let Z be a discrete-time normal noise that has the chaotic representation property. We first prove a result concerning the regularity of generalized functionals of Z. Then, we use the Fock transform to define some fundamental operators on generalized functionals of Z and apply the abovementioned regularity result to prove the continuity of these operators. Finally, we establish the Clark-Ocone formula for generalized functionals of Z and show its application results, which include the covariant identity result and the variant upper bound result for generalized functionals of Z.

Replicating portfolio approach to capital calculation

The replicating portfolio (RP) approach to the calculation of capital for life insurance portfolios is an industry standard. The RP is obtained from projecting the terminal loss of discounted asset–liability cash flows on a set of factors generated by a family of financial instruments that can be efficiently simulated. We provide the mathematical foundations and a novel dynamic and path-dependent RP approach for real-world and risk-neutral sampling. We show that our RP approach yields asymptotically consistent capital estimators if the chaotic representation property holds. We illustrate the tractability of the RP approach by three numerical examples.

A characterization of operators on functionals of discrete-time normal martingales

ABSTRACTIn this article, we aim at characterizing operators acting on functionals of discrete-time normal martingales. Let be a discrete-time normal martingale that has the chaotic representation property. We first introduce a transform, called 2D-Fock transform, for operators from the testing functional space to the generalized functional space of M. Then we characterize continuous linear operators from to via their 2D-Fock transforms. Our characterization theorems show that there exists a one-to-one correspondence between continuous linear operators from to and functions on Γ × Γ that only satisfy some type of growth condition, where Γ designates the finite power set of . Finally, we give some applications of our characterization theorems.

Operators of stochastic differentiation on spaces of nonregular generalized functions of Levy white noise analysis

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 some 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, in the same way as on spaces of regular test and generalized functions and on spaces of nonregular test functions of the Levy white noise analysis. In the present paper we make the next natural step: introduce and study operators of stochastic differentiation on spaces of nonregular generalized functions of the Levy white noise analysis (i.e., on spaces of generalized 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). In so doing, we use Lytvynov's generalization of the chaotic representation property. The researches of the present paper can be considered as a contribution in a further development of the Levy white noise analysis.

Replicating Portfolio Approach to Capital Calculation

The replicating portfolio (RP) approach to the calculation of capital for life insurance portfolios is an industry standard. The RP is obtained from projecting the terminal loss of discounted asset-liability cash flows on a set of factors generated by a family of financial instruments that can be efficiently simulated. We provide the mathematical foundations and a novel dynamic and path-dependent RP approach for real-world and risk-neutral sampling. We show that our RP approach yields asymptotically consistent capital estimators if the chaotic representation property holds. We illustrate the tractability of the RP approach by two numerical examples.

The chaotic representation property of compensated-covariation stable families of martingales

In the present paper, we study the chaotic representation property for certain families XX of square integrable martingales on a finite time interval [0,T][0,T]. For this purpose, we introduce the notion of compensated-covariation stability of such families. The chaotic representation property will be defined using iterated integrals with respect to a given family XX of square integrable martingales having deterministic mutual predictable covariation ⟨X,Y⟩⟨X,Y⟩ for all X,Y∈XX,Y∈X. The main result of the present paper is stated in Theorem 5.8 below: If XX is a compensated-covariation stable family of square integrable martingales such that ⟨X,Y⟩⟨X,Y⟩ is deterministic for all X,Y∈XX,Y∈X and, furthermore, the system of monomials generated by XX is total in L2(Ω,FXT,P)L2(Ω,FTX,P), then XX possesses the chaotic representation property with respect to the σσ-field FXTFTX. We shall apply this result to the case of Levy processes. Relative to the filtration FLFL generated by a Levy process LL, we construct families of martingales which possess the chaotic representation property. As an illustration of the general results, we will also discuss applications to continuous Gaussian families of martingales and independent families of compensated Poisson processes. We conclude the paper by giving, for the case of Levy processes, several examples of concrete families XX of martingales including Teugels martingales.

Characterization Theorems for Generalized Functionals of Discrete-Time Normal Martingale

We aim at characterizing generalized functionals of discrete-time normal martingales. LetM=(Mn)n∈Nbe a discrete-time normal martingale that has the chaotic representation property. We first construct testing and generalized functionals ofMwith an appropriate orthonormal basis forM’s square integrable functionals. Then we introduce a transform, called the Fock transform, for these functionals and characterize them via the transform. Several characterization theorems are established. Finally we give some applications of these characterization theorems. Our results show that generalized functionals of discrete-time normal martingales can be characterized only by growth condition, which contrasts sharply with the case of some continuous-time processes (e.g., Brownian motion), where both growth condition and analyticity condition are needed to characterize generalized functionals of those continuous-time processes.

On operators of stochastic differentiation on spaces of regular test and generalized functions of Lévy white noise analysis

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. In this paper we introduce and study bounded and unbounded operators of stochastic differentiation in the Levy white noise analysis. More exactly, we consider these operators on spaces from parametrized regular rigging of the space of square integrable with respect to the measure of a Levy white noise functions, using the Lytvynov's generalization of the chaotic representation property. This gives a possibility to extend to the Levy white noise analysis and to deepen the corresponding results of the classical white noise analysis.

On extended stochastic integrals with respect to Lévy processes

Let $L$ be a Levy process on $[0,+\infty)$. 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 Levy process the CRP does not hold.
 There are different generalizations of the CRP for Levy processes. In particular, under the Ito's approach one decomposes a Levy 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 cadlag version of $L$. The Lytvynov's approach is based on orthogonalization of continuous monomials in the space of square integrable random variables.
 In this paper we construct the extended stochastic integral with respect to a Levy 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.

COHERENT STATES IN BERNOULLI NOISE FUNCTIONALS

AbstractLet (Ω,ℱ,ℙ) be a probability space and Z=(ZK)k∈ℕ a Bernoulli noise on (Ω,ℱ,ℙ) which has the chaotic representation property. In this paper, we investigate a special family of functionals of Z, which we call the coherent states. First, with the help of Z, we construct a mapping ϕ from l2(ℕ) to ℒ2(Ω,ℱ,ℙ) which is called the coherent mapping. We prove that ϕ has the continuity property and other properties of operation. We then define functionals of the form ϕ(f) with f∈l2 (ℕ) as the coherent states and prove that all the coherent states are total in ℒ2 (Ω,ℱ,ℙ) . We also show that ϕ can be used to factorize ℒ2 (Ω,ℱ,ℙ) . Finally we give an application of the coherent states to calculus of quantum Bernoulli noise.

The explicit chaotic representation of the powers of increments of Lévy processes

This paper presents a computationally explicit formula of the chaotic representation property (CRP) for the powers of increments of a Lévy process. The formula can be used to obtain the integrands of the CRP in terms of orthogonal compensated power jump processes and the CRP in terms of Poisson random measures. Simulation results demonstrate that the performance of the representation is satisfactory. The CRP of a number of financial derivatives can be found by expressing them in terms of the powers of increments of the underlying Lévy process using Taylor's expansion.

Hermite and Laguerre Polynomials and Matrix-Valued Stochastic Processes

We extend to matrix-valued stochastic processes, some well-known relations between real-valued diffusions and classical orthogonal polynomials, along with some recent results about Levy processes and martingale polynomials. In particular, joint semigroup densities of the eigenvalue processes of the generalized matrix-valued Ornstein-Uhlenbeck and squared Ornstein-Uhlenbeck processes are respectively expressed by means of the Hermite and Laguerre polynomials of matrix arguments. These polynomials also define martingales for the Brownian matrix and the generalized Gamma process. As an application, we derive a chaotic representation property for the eigenvalue process of the Brownian matrix.

The chaotic-representation property for a class of normal martingales

Suppose \(Z=(Z_t)_{t\ge0}\) is a normal martingale which satisfies the structure equation $$d[Z]_t = (\alpha(t)+\beta(t)Z_{t-}) dZ_t + dt$$ . By adapting and extending techniques due to Parthasarathy and to Kurtz, it is shown that, if α is locally bounded and β has values in the interval [-2,0], the process Z is unique in law, possesses the chaotic-representation property and is strongly Markovian (in an appropriate sense). If also β is bounded away from the endpoints 0 and 2 on every compact subinterval of [0,∞] then Z is shown to have locally bounded trajectories, a variation on a result of Russo and Vallois.