Normal Martingale 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.

BSDEs driven by normal martingale

The aim of this paper is to study backward stochastic differential equation driven by a class of normal martingale. We prove the existence and uniqueness results of the solution under the stochastic Lipschitz condition by mean the martingale representation theorem. In addition, we show that the solution of these equations provides a viscosity solution of the associated system with partial differential equations.

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 integral equations of Volterra type in terms of discrete-time normal martingale

In this paper, we aim to introduce a quantum linear stochastic Volterra integral equation of convolution type with operator-valued kernels in a nuclear topological algebra. We first establish the existence and uniqueness of the solutions and give the explicit expression of the solutions. Then we prove the continuity, continuous dependence on free terms and other properties of the solution.

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.

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.

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.

CONVOLUTION OF FUNCTIONALS OF DISCRETE-TIME NORMAL MARTINGALES

AbstractLet M=(M)n∈ℕ be a discrete-time normal martingale satisfying some mild requirements. In this paper we show that through the full Wiener integral introduced by Wang et al. (‘An alternative approach to Privault’s discrete-time chaotic calculus’, J. Math. Anal. Appl.373 (2011), 643–654), one can define a multiplication-type operation on square integrable functionals of M, which we call the convolution. We examine algebraic and analytical properties of the convolution and, in particular, we prove that the convolution can be used to represent a certain family of conditional expectation operators associated with M. We also present an example of a discrete-time normal martingale to show that the corresponding convolution has an integral representation.

On the path structure of a semimartingale arising from monotone probability theory

Let $X$ be the unique normal martingale such that $X_0=0$ and \[\mathrm{d}[X]_t=(1-t-X_{t-}) \mathrm{d}X_t+\mathrm{d}t\] and let $Y_t:=X_t+t$ for all $t\geq 0$; the semimartingale $Y$ arises in quantum probability, where it is the monotone-independent analogue of the Poisson process. The trajectories of $Y$ are examined and various probabilistic properties are derived; in particular, the level set $\{t\geq 0\dvt Y_t=1\}$ is shown to be non-empty, compact, perfect and of zero Lebesgue measure. The local times of $Y$ are found to be trivial except for that at level 1; consequently, the jumps of $Y$ are not locally summable.

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.

Quantum stopping times and quasi-left continuity

We study the general properties of quantum stopping times on Hilbert spaces equipped with a filtration. We define and investigate notions such as the spaces of anterior events, the spaces of strictly anterior events and above all we define the property S< T for two stopping times together with the notion of predictable quantum stopping times. It is well-known that the natural filtration of any normal martingale with the predictable representation property is quasi-left continuous; with the help of our new notions we prove that this property is actually an intrinsic property of the symmetric Fock space Φ over L 2( R +) . We also apply these definitions to the case of a non commutative stochastic base. We show, in this context, that the fermionic Fock space over L 2( R +) , the quasi-free boson and fermion spaces are also quasi-left continuous.

Anticipating stratonovich integral with respect to the Azema's martingales

In this paper we establish the conditions on a L 2-process u for the existence of its anticipating Stratonovich integral with respect to a normal martingale belonging to a certain class. This class includes the Azema's martingales and the compensated Poisson processes.

Product of two multiple stochastic integrals with respect to a normal martingale

Let M be a normal martingale (i.e. 〈 M, M〉 ( t) = t), we decompose the product of two multiple stochastic integrals (with respect to M) I n ( f) I m ( g) as a sum of n ∧ m terms H k . H k is equal to the integral over R k + of the function t → I n+ m−2 k ( h k ( t,.)), with respect to the k-tensor product of d[ M, M]., h k being an explicit function depending only on f and g. Our formula generalizes the well-known result concerning Brownian motion and compensated Poisson process and allows us to improve some results of Emery related to the chaos representation property of solution of the structure equation.