Theitô Formula for Quantum Semimartingales

Vincent-Smith, G. F.

doi:10.1112/s0024611597000476

Cited by 11 publications

(19 citation statements)

References 18 publications

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“…We follow the proof of [16,Theorem 4.1], with appropriate modifications. It is easy to verify that the coefficients given are W-adapted and satisfy the requisite integrability conditions (e.g., both r l (M(t)) and r l (P(t)) are bounded in norm by (R − R 0 ) −1 , uniformly in l ¥ c and t ¥ [0, y]).…”

Section: Theorem 22 If M Is a W-semimartingale And F Is A Function mentioning

confidence: 99%

“…We let S W denote the algebra formed by these integrals; it is analogous to the algebra of regular quantum semimartingales studied by Attal [2]. We produce polynomial and holomorphic-functional Itô formulae for elements of S W , in the manner of Vincent-Smith [16]. The fact that W-semimartingales (i.e., elements of S W ) have well-behaved norm allows us to solve the modification of the evolution equation appropriate to W-adapted processes; in the usual notation of quantum stochastic calculus this is…”

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Quantum Ω-Semimartingales and Stochastic Evolutions

Belton¹

2001

Journal of Functional Analysis

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We explore W-adaptedness, a variant of the usual notion of adaptedness found in stochastic calculus. It is shown that the (non-adapted) quantum stochastic integrals of bounded, W-adapted processes are themselves bounded and W-adapted, a fact that may be deduced from the Bismut-Clark-Ocone formula of Malliavin calculus. An algebra analogous to Attal's class S of regular quantum semimartingales is defined, and product and functional Itô formulae are given. We consider quantum stochastic differential equations with bounded, W-adapted coefficients that are time dependent and act on the whole Fock space. Solutions to such equations may be used to dilate quantum dynamical semigroups in a manner that generalises, and gives new insight into, that of R. Alicki and M. Fannes (1987, Comm. Math. Phys. 108, 353-361); their unitarity condition is seen to be the usual condition of R. L. Hudson and K. R. Parthasarathy (1984, Comm. Math. Phys 93, 301-323). © 2001 Elsevier ScienceKey Words: W-adaptedness; quantum semimartingales; quantum stochastic differential equations; quantum dynamical semigroups. INTRODUCTIONIn 1987 Alicki and Fannes [1] demonstrated a method of dilating quantum dynamical semigroups using classical Brownian motion. They solved the vector equation), where B = (B 1 , ..., B d ) is a standard d-dimensional Wiener process on W (d) , E is the conditional expectation with respect to the filtration generated by this process, This leads to a unitary process which dilates the quantum dynamical semigroup on B(H) with generatorVincent-Smith [15] noted that their technique sits naturally within (a non-adapted extension of) Hudson and Parthasarathy's quantum stochastic calculus [10] and proved the following. The equation (1) is equivalent (in one dimension) to the (non-adapted) quantum stochastic equation) and H, Boson Fock space over L 2 [0, .). If we take V and K to be adapted processes of operators with essentially bounded norm, acting on H é H and satisfying Eq. (2) almost everywhere, then there is a unique, unitary solution W.When Lindsay [12] developed the non-adapted QS calculus he observed that it is directly applicable to the situation considered by Vincent-Smith, and he coined the term W-adapted, pronounced ''vacuum-adapted,'' to describe the class of integrands: they are the composition of adapted processes (in the sense of Hudson and Parthasarathy) and the conditional expectation process. Such processes have been investigated by Hudson and Krée [9], who demonstrated the link between one-dimensional quantum stochastic calculus with Hilbert-Schmidt, W-adapted processes and two-dimensional classical Itô calculus. Belavkin [5] has also considered integrals of W-adapted processes in his generalised definition of non-adapted QS integrals, which uses a scale of Fock spaces, and provides a quantum Itô formula for such integrals. QUANTUM W-SEMIMARTINGALES 95In the first part of this paper we study bounded, W-adapted processes. It is remarkable that the quantum stochastic integrals of such processes are themsel...

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Section: Theorem 22 If M Is a W-semimartingale And F Is A Function mentioning

confidence: 99%

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confidence: 99%

Quantum Ω-Semimartingales and Stochastic Evolutions

Belton¹

2001

Journal of Functional Analysis

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“…This key fact, together with the classical theory of unbounded self-adjoint operators in Hilbert space, allows the extension to M of the functional quantum Ito formula (see [10,Theorem 6.2] and [2,Theorem I 18]). …”

Section: Introductionmentioning

confidence: 99%

The Ito Formula for Perturbed Brownian Martingales

Vincent-Smith¹

2000

Bulletin of the London Mathematical Society

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“…In this Itô calculus formulation operator composition of QS integrals is admitted. Under some conditions the domain of these QS integrals may be the whole of Fock space-a fact which plays an important role in the theory of quantum semimartingale algebras and the theory of quantum square and angle brackets [2,43]. The main disadvantage of this formulation is that the QS integrals are only defined implicitly.…”

Section: Introduction Quantum Stochastic Calculus Is Now a Well-estamentioning

confidence: 99%

“…[11,25]. However, to date, the most developed [2,5,9,16,17,19,22,27,29,30,34,33,38,43] is the bosonic theory originated by Hudson and Parthasarathy [23]. Moreover, fermionic theory has been incorporated into the bosonic by means of a continuous Jordan-Wigner transformation which has a simple quantum stochastic description [24].…”

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Quantum stochastic calculus with maximal operator domains

Attal¹,

Lindsay²

2004

Ann. Probab.

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Dedicated to Robin Hudson in the year of his 60th birthdayQuantum stochastic calculus is extended in a new formulation in which its stochastic integrals achieve their natural and maximal domains. Operator adaptedness, conditional expectations and stochastic integrals are all defined simply in terms of the orthogonal projections of the time filtration of Fock space, together with sections of the adapted gradient operator. Free from exponential vector domains, our stochastic integrals may be satisfactorily composed yielding quantum Itô formulas for operator products as sums of stochastic integrals. The calculus has seen two reformulations since its discovery-one closely related to classical Itô calculus; the other to noncausal stochastic analysis and Malliavin calculus. Our theory extends both of these approaches and may be viewed as a synthesis of the two. The main application given here is existence and uniqueness for the Attal-Meyer equations for implicit definition of quantum stochastic integrals. [11,25]. However, to date, the most developed [2,5,9,16,17,19,22,27,29,30,34,33,38,43] is the bosonic theory originated by Hudson and Parthasarathy [23]. Moreover, fermionic theory has been incorporated into the bosonic by means of a continuous Jordan-Wigner transformation which has a simple quantum stochastic description [24]. Introduction.As originally set down the homogeneity of exponential vectors with respect to the continuous tensor product structure of (symmetric) Fock space was key to the development of the calculus. Quantum stochastic (QS) integrals were constructed for time-indexed families of operators which are defined on exponential vectors and satisfy an adaptedness property which is itself nicely described in terms of these vectors. The QS integrals were thereby defined on such exponential domains, too. Their composition as operators is thus inadmissible, strictly speaking, within this exponential vector formulation, since they typically do not leave the linear

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Theitô Formula for Quantum Semimartingales

Cited by 11 publications

References 18 publications

Quantum Ω-Semimartingales and Stochastic Evolutions

Quantum Ω-Semimartingales and Stochastic Evolutions

The Ito Formula for Perturbed Brownian Martingales

Quantum stochastic calculus with maximal operator domains

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