Normal Martingales Research Articles

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.

The Bochner-convolution integral for generalized functional-valued functions of discrete-time normal martingales

Let M be a discrete-time normal martingale satisfying some mild conditions, S M ⊂ L 2 M ⊂ S∗ M be the Gel’fand triple constructed from the functionals of M . As is known, there is no usual multiplication in S ∗ M since its elements are continuous linear functionals on S M . However, by using the Fock transform, one can introduce convolution in S ∗ M , which suggests that one can try to introduce a type of integral of an S ∗ M -valued function with respect to an S ∗ M -valued measure in the sense of convolution. In this paper, we just define such type of an integral. First, we introduce a class of S ∗ M -valued measures and examine their basic properties. Then, we define an integral of an S ∗ M -valued function with respect to an S ∗ M -valued measure and, among others, we establish a dominated convergence theorem for this integral. Finally, we also prove a Fubini type theorem for this integral.

Quantum Stochastic Cable Equation Acting on Functionals of Discrete-Time Normal Martingales

Let M be a discrete-time normal martingale satisfying some mild conditions. Then Gel’fand triple S(M)⊂L2(M)⊂S⁎(M) can be constructed of functionals of M, where elements of S(M) are called testing functionals of M, while elements of S⁎(M) are called generalized functionals of M. In this paper, we consider a quantum stochastic cable equation in terms of operators from S(M) to S⁎(M). Mainly with the 2D-Fock transform as the tool, we establish the existence and uniqueness of a solution to the equation. We also examine the continuity of the solution and its continuous dependence on initial values.

Bounds in Total Variation Distance for Discrete-time Processes on the Sequence Space

Let $\mathbb {P}$ and $\widetilde {\mathbb {P}}$ be the laws of two discrete-time stochastic processes defined on the sequence space $S^{\mathbb N}$, where S is a finite set of points. In this paper we derive a bound on the total variation distance $\mathrm {d}_{\text {TV}}(\mathbb {P},\widetilde {\mathbb {P}})$ in terms of the cylindrical projections of $\mathbb {P}$ and $\widetilde {\mathbb {P}}$. We apply the result to Markov chains with finite state space and random walks on $\mathbb {Z}$ with not necessarily independent increments, and we consider several examples. Our approach relies on the general framework of stochastic analysis for discrete-time obtuse random walks and the proof of our main result makes use of the predictable representation of multidimensional normal martingales. Along the way, we obtain a sufficient condition for the absolute continuity of $\widetilde {\mathbb {P}}$ with respect to $\mathbb {P}$ which is of interest in its own right.

Convergence Theorems for Operators Sequences on Functionals of Discrete-Time Normal Martingales

We aim to investigate the convergence of operators sequences acting on functionals of discrete-time normal martingales M. We first apply the 2D-Fock transform for operators from the testing functional space S(M) to the generalized functional space S⁎(M) and obtain a necessary and sufficient condition for such operators sequences to be strongly convergent. We then discuss the integration of these operator-valued functions. Finally, we apply the results obtained here and establish the existence and uniqueness of solution to quantum stochastic differential equations in terms of operators acting on functionals of discrete-time normal martingales M. And also we prove the continuity and continuous dependence on initial values of the solution.

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.

Complex obtuse random walks and their continuous-time limits

We study a particular class of complex-valued random variables and their associated random walks: the complex obtuse random variables. They are the generalization to the complex case of the real-valued obtuse random variables which were introduced in \cite{A-E} in order to understand the structure of normal martingales in $\RR^n$.The extension to the complex case is mainly motivated by considerations from Quantum Statistical Mechanics, in particular for the seek of a characterization of those quantum baths acting as classical noises. The extension of obtuse random variables to the complex case is far from obvious and hides very interesting algebraical structures. We show that complex obtuse random variables are characterized by a 3-tensor which admits certain symmetries which we show to be the exact 3-tensor analogue of the normal character for 2-tensors (i.e. matrices), that is, a necessary and sufficient condition for being diagonalizable in some orthonormal basis. We discuss the passage to the continuous-time limit for these random walks and show that they converge in distribution to normal martingales in $\CC^N$. We show that the 3-tensor associated to these normal martingales encodes their behavior, in particular the diagonalization directions of the 3-tensor indicate the directions of the space where the martingale behaves like a diffusion and those where it behaves like a Poisson process. We finally prove the convergence, in the continuous-time limit, of the corresponding multiplication operators on the canonical Fock space, with an explicit expression in terms of the associated 3-tensor again.

Convergence Theorems for Generalized Functional Sequences of Discrete-Time Normal Martingales

The Fock transform recently introduced by the authors in a previous paper is applied to investigate convergence of generalized functional sequences of a discrete-time normal martingaleM. A necessary and sufficient condition in terms of the Fock transform is obtained for such a sequence to be strongly convergent. A type of generalized martingales associated withMis introduced and their convergence theorems are established. Some applications are also shown.

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.

Stochastic Analysis for Obtuse Random Walks

We present a construction of the basic operators of stochastic analysis (gradient and divergence) for a class of discrete-time normal martingales called obtuse random walks. The approach is based on the chaos representation property and discrete multiple stochastic integrals. We show that these operators satisfy similar identities as in the case of the Bernoulli randoms walks. We prove a Clark-Ocone-type predictable representation formula, obtain two covariance identities and derive a deviation inequality. We close the exposition by an application to option hedging in discrete time.

Chaotic extensions and the lent particle method for Brownian motion

In previous works, we have developed a new Malliavin calculus on the Poisson space based on the lent particle formula. The aim of this work is to prove that, on the Wiener space for the standard Ornstein-Uhlenbeck structure, we also have such a formula which permits to calculate easily and intuitively the Malliavin derivative of a functional. Our approach uses chaos extensions associated to stationary processes of rotations of normal martingales.

HILBERT SPACE-VALUED INTEGRAL OF OPERATOR-VALUED FUNCTIONS

In this paper we construct and study an integral of operator-valued functions with respect to Hilbert space-valued measures generated by a resolution of identity. Our integral generalizes the Itô stochastic integral with respect to normal martingales and the Itô integral on a Fock space.

Stochastic control problems for systems driven by normal martingales

In this paper we study a class of stochastic control problems in which the control of the jump size is essential. Such a model is a generalized version for various applied problems ranging from optimal reinsurance selections for general insurance models to queueing theory. The main novel point of such a control problem is that by changing the jump size of the system, one essentially changes the type of the driving martingale. Such a feature does not seem to have been investigated in any existing stochastic control literature. We shall first provide a rigorous theoretical foundation for the control problem by establishing an existence result for the multidimensional structure equation on a Wiener–Poisson space, given an arbitrary bounded jump size control process; and by providing an auxiliary counterexample showing the nonuniqueness for such solutions. Based on these theoretical results, we then formulate the control problem and prove the Bellman principle, and derive the corresponding Hamilton–Jacobi–Bellman (HJB) equation, which in this case is a mixed second-order partial differential/difference equation. Finally, we prove a uniqueness result for the viscosity solution of such an HJB equation.

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.

Chaotic representation property of certain Azéma martingales

An open problem in martingale theory is to characterize the normal martingales having the chaotic representation property. This property is shown to hold for certain families of multidimensional Azema martingales.

From [formula omitted]-level atom chains to n-dimensional noises

In quantum physics, the state space of a countable chain of ( n + 1 ) -level atoms becomes, in the continuous field limit, a Fock space with multiplicity n. In a more functional analytic language, the continuous tensor product space over R + of copies of the space C n + 1 is the symmetric Fock space Γ s ( L 2 ( R + ; C n ) ) . In this article we focus on the probabilistic interpretations of these facts. We show that they correspond to the approximation of the n-dimensional normal martingales by means of obtuse random walks, that is, extremal random walks in R n whose jumps take exactly n + 1 different values. We show that these probabilistic approximations are carried by the convergence of the matrix basis a j i ( p ) of ⊗ N C n + 1 to the usual creation, annihilation and gauge processes on the Fock space.

Umbral Calculus, Martingales, and Associated Polynomials

In this paper we treat with a relationship between normal martingales [Emery, M. On the Azéma Martingales; Sem. Proba. XXIII, Lec. Notes in Maths 1372; Springer, 1989; 66–87] and some special polynomial families. This problem was studied by Privault et al. [Privault, N.; Sole, J.L.; Vives, J. Chaotic Kabanov Formula for the Azéma Martingales; The Bernoulli Society for Mathematical Statistics and Probability, 2000; Vol. 6, No. 4], they showed under some assumptions which we give later, that Wiener and compensated Poisson processes are the only normal martingales which has an associated family of polynomials according to the Definition 3 in Privault et al. [Privault, N.; Sole, J.L.; Vives, J. Chaotic Kabanov Formula for the Azéma Martingales; The Bernoulli Society for Mathematical Statistics and Probability, 2000; Vol. 6, No. 4]. We study this problem without the authors assumptions in considering all normal martingales, we show more precisely that Wiener process is characterized by Wick polynomials and both Wiener and compensated Poisson processes are characterized by cross polynomials (said also of convolution type). We use here the Umbral Calculus theory [Roman, S.M. The theory of umbral calculus I. J. Math. Anal. Appl. 1982, 87, 58–115] as a new and a fast tool to solve a martingale problem.

Normal martingales and polynomial families

Normal martingales and polynomial families

Hedging in complete markets driven by normal martingales

This paper aims at a unied treatment of hedging in market models driven by martingales with deterministic brackethM;Mit, including Brownian motion and the Poisson process as particular cases. Replicating hedging strategies for European, Asian and Lookback options are explicitly computed using either the Clark{Ocone formula or an extension of the delta hedging method, depending on which is most appropriate.

Extended covariance identities and inequalities

We state an abstract version of covariance identities and inequalities for normal martingales, which uses any gradient operator that satisfies a Clark formula. This extends and makes more precise some results of Houdré and Pérez-Abreu (Ann. Probab. 23 (1995)), with simplified proofs.