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 normalmartingalesM.We first apply the 2D-Fock transform for operators from the testing functional spaceS(M) to the generalized functional spaceS∗(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.


Introduction
White noise analysis created by Hida [1] is essentially a branch of infinite-dimensional calculus on generalized functionals of Brownian motion, which connected with the applications to the study of random processes and stochastic differential equations.Ito's theory is based on the notion of the Wiener measure in the space of continuous functions, interpreted as the trajectories of Brownian motion.And considering the space of slowly growing Schwartz distributions as the space of trajectories of white noise, Hida has constructed a new theory of white noise functionals [2][3][4], which proved to be useful in the studying of stochastic differential equations and was further developed in [5,6] and other papers.On the other hand, many problems in mathematical physics and financial mathematics lead to the necessity of studying stochastic differential equations with operator coefficients in infinitedimensional spaces.In [7], the authors generalize the classical Ito theory to this case.
In 1998, Ito introduced his analysis of generalized Poisson functionals in paper [8], which can be viewed as a theory of infinite-dimensional calculus on generalized functionals of Poisson martingale.As we know, both Brownian motion and Poisson martingale are continuous-time normal martingales.And other continuous-time processes theories of white noise analysis appeared in [9][10][11][12].
Discrete-time normal martingales [13] also play an important role in many theoretical and applied fields.For example, the classical random walk is just such a discretetime normal martingale [14,15].It would then be interesting to develop a theory of infinite-dimensional calculus on generalized functionals of discrete-time normal martingales.
Let  = (  ) ∈N be a discrete-time normal martingale satisfying some mild conditions.In paper [16], the authors constructed the testing functional space S() and generalized functional space S * () of  by using a specific orthonormal basis for square integrable functionals of  and also characterized these functionals via a transform acting on them.
It is well known that operators on functional spaces play a fundamental role in quantum mechanics.In paper [17], the authors introduce a transform, called 2D-Fock transform, for operators from the testing functional space S() to the generalized functional space S * () of  and characterize continuous linear operators from S() to S * ().In this paper, we apply the 2D-Fock transform to investigate the convergence of operators sequences acting on functionals of discrete-time normal martingales .We obtain a necessary and sufficient condition for such sequence to be strongly convergent.And we give a criterion for checking whether such operator-valued functions are Bochner-integrable.
Notation and Conventions.Throughout the paper, N designates the set of all nonnegative integers and Γ denotes the finite power set of N; namely, where #() means the cardinality of  as a set.In addition, we always assume that (Ω, F, ) is a given probability space with E denoting the expectation with respect to .We denote by  2 (Ω, F, ) the usual Hilbert space of square integrable complex-valued functions on (Ω, F, ) and use ⟨⋅, ⋅⟩ and ‖ ⋅ ‖ to mean its inner product and norm, respectively.By convention, ⟨⋅, ⋅⟩ is conjugate-linear in its first argument and linear in its second argument.
Let  = (  ) ∈N be a discrete-time normal martingale on (Ω, F, ).Then one can construct from  a process  = (  ) ∈N as It can be verified that  admits the following properties: Thus  can be viewed as a discrete-time noise, which we call the discrete-time normal noise associated with .
Let F ∞ = (  ;  ∈ N); the -field over Ω is generated by .In the literature, F ∞ -measurable functions on Ω are also known as functionals of .Thus elements of  2 (Ω, F ∞ , ) can be called square integrable functionals of .Definition 3. The discrete-time normal martingale  is said to have the chaotic representation property if the system {  |  ∈ Γ} defined by ( 5) is total in  2 (Ω, F ∞ , ).
So, if the discrete-time normal martingale  has the chaotic representation property, then the system {  |  ∈ Γ} defined by ( 5) is actually an orthonormal basis for  2 (Ω, F ∞ , ), which is a closed subspace of  2 (Ω, F, ) as is known.
Lemma 4 (see [20]).Let   →   be the N-valued function on Γ given by Then, for  > 1, the positive term series ∑ ∈Γ  −  converges and moreover Using the N-valued function defined by (7), we can construct a chain of Hilbert spaces consisting of functionals of  as follows.For  ≥ 0, we define a norm ‖ ⋅ ‖  on and put It is not hard to check that ‖ ⋅ ‖  is a Hilbert norm and S  () becomes a Hilbert space with ‖ ⋅ ‖  .
It is easy to see that   ≥ 1 for all  ∈ Γ.This implies that ‖ ⋅ ‖  ≤ ‖ ⋅ ‖  and S  () ⊂ S  () whenever 0 ≤  ≤ .Thus we actually get a chain of Hilbert spaces of functionals of : We now put and endow it with the topology generated by the norm sequence {‖ ⋅ ‖  } ≥0 .Note that, for each  ≥ 0, S  () is just the completion of S() with respect to ‖ ⋅ ‖  .Thus S() is a countably-Hilbert space [21].The next lemma, however, shows that S() even has a much better property.
Lemma 7 (see [16]).Let S * () be the dual of S() and endow it with the strong topology.Then and moreover the inductive limit topology on S * () given by space sequence {S *  ()} ≥0 coincides with the strong topology.
We mention that, by identifying  2 () with its dual, one comes to a Gel'fand triple, which we refer to as the Gel'fand triple associated with .
Definition 8 (see [16]).Elements of S * () are called generalized functionals of , while elements of S() are called testing functionals of .
Throughout this paper, we denote by L the set of all continuous linear operators from S() to S * (); that is, Definition 9 (see [17]).For an operator  ∈ L, its 2D-Fock transform is the function T on Γ × Γ given by where ⟨⟨⋅, ⋅⟩⟩ is the canonical bilinear form on S * () × S().
Much like generalized functionals of , continuous linear operators in L are also completely determined by their 2D-Fock transforms.
The following lemma is known as the characterization theorem of operators in L through their 2D-Fock transforms.
Lemma 11 (see [17]).Let  be a function on Γ × Γ.Then  is the 2D-Fock transform of an element  in L if and only if it satisfies for some constants  ≥ 0 and  ≥ 0. In that case, for  >  + 1/2, one has and in particular  takes values in S *  (), where where  ≥ 0.
For two operators  1 ,  2 ∈ L, their usual product  1  2 may not make sense.However, one can introduce a product of other type for them.

Convergence Theorems for Operators in L
Let  = (  ) ∈N be the same discrete-time normal martingale as described in Section 2. In the present section, we apply the 2D-Fock transform to establish convergence theorems for operators in L. Furthermore, we discuss the integration of these operator-valued functions.Definition 13.A sequence (  ) ≥1 ⊂ L is called strongly convergent to  ∈ L, if, for any  ∈ S(), one has   () → () (in the strong topology of S * ()).
The following theorem offers a necessary condition for operators sequences in L to be strongly convergent.
In the following, we discuss the integration of L-valued mapping.
Definition 15.Let (, E, ]) be a measure space and let (⋅) :  → L be a L-valued function; then (⋅) is said to be strongly Bochner-integrable with respect to ] if generalized functional valued function (⋅) :  → S * () is Bochnerintegrable with respect to ] for any  ∈ S().In that case, a linear operator from S() to S * (), is defined.We called it the Bochner integral of (⋅) with respect to ] and it is denoted by ∫  ()]().
We note that the strong Bochner integral ∫  ()]() may not be an operator in L. The next theorem, however, provides sufficient conditions for (⋅) to be strongly Bochnerintegrable and its strong Bochner integral to be an operator in L.

Application to Quantum SDEs
In the present section, we show applications of our main results in Section 3. We establish the existence and uniqueness of solution to (1).Equation ( 1) can describe quantum stochastic evolutions in the presence of quantum noise.As usual, by a solution of (1) we mean a map  : R + → L which satisfies the following integral equation: where the integral is the Bochner integral of operator function in L. (1)  : R + → L is continuous.
In the following, we always assume that the map ,  : R + × L → L and the noise process (()) ∈R + are the same as in Theorem 20.For  ∈ L, we denote by {(, ) |  ∈ R + } the solutions of (1) with  being its initial value.The following theorem shows that the solution to (1) continuously depends on the initial value.(60) Proof.For 0 ≤  < +∞, let  ≥ 0,  > 0, and  ≥ 1 be such that, for all  ∈ [0, ] and ,  ∈ Γ, we have It is easy to see that, for any  ∈ L,
Definition 19.Let Θ : R + × L → L be a map.Θ is said to satisfy a locally uniform Lipschitz condition and a locally uniform linear growth condition, if, for any  ≥ 0, there exists a constant  ≥ 0, such that, for any ,  ∈ Γ and ,  ∈ L, one Let ,  : R + × L → L be continuous and satisfy a locally uniform Lipschitz condition and locally uniform linear growth condition, and let the noise process (()) ∈R + satisfy the following: = +e  + e  .Thus, it follows from Theorem 14 that for any  ∈ [0, ], (  ()) ≥1 converges in L.