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Journal of Mathematical Physics

The unified Ito formula has the pseudo-Poisson structure df(x)=[f(x+b)−f(x)]μνdaνμ

V. P. Belavkin

Published Apr 1, 1993• 6 citations
Publisher

Abstract

A universal functional Ito formula, recently discovered within the *-Ito noncommutative algebraic approach, is described herein. The formula unifies the classical Ito formulas for Wiener, Poisson, and quantum cases, extends the chain rule df(x1,...,xl) = ∑f′i(x)dxi to the case of noncommuting stochastically differentiable operators Xi(t), and also admits a nonadapted generalization.

DOI
10.1063/1.530170

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A universal functional Ito formula, recently discovered within the *-Ito noncommutative algebraic approach, is described herein. The formula unifies the classical Ito formulas for Wiener, Poisson, and quantum cases, extends the chain rule df(x1,...,xl) = ∑f′i(x)dxi to the case of noncommuting stochastically differentiable operators Xi(t), and also admits a nonadapted generalization.
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A simple axiomatic characterization of the general (infinite dimensional, noncommutative) Ito algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. The notion of Ito B*-algebra, generalizing the C*-algebra is defined to include the Banach infinite dimensional Ito algebras of quantum Brownian and quantum Levy motion, and the B*-algebras of vacuum and thermal quantum noise are characterized. It is proved that every Ito algebra is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Levy (Poisson) algebra. In particular, every quantum thermal noise is the orthogonal sum of a quantum Wiener noise and a quantum Poisson noise as it is stated by the Levy-Khinchin Theorem in the classical case .
We review the basic concepts of quantum probability and stochastics using the universal Itô B *-algebra approach. The main notions and results of classical and quantum stochastics are reformulated in this unifying approach. The general Lévy process is defined in terms of the modular B *-Itô algebra, and the corresponding quantum stochastic master equation on the predual space of the W *-algebra is derived as a noncommutative version of the Zakai equation driven by the process. This is done by a noncommutative analog of the Girsanov transformation, which we introduce here in full generality.
A simple axiomatic characterization of the general (infinite dimensional, noncommutative) Ito algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. The notion of Ito B*-algebra, generalizing the C*-algebra is defined to include the Banach infinite dimensional Ito algebras of quantum Brownian and quantum Levy motion, and the B*-algebras of vacuum and thermal quantum noise are characterized. It is proved that every Ito algebra is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Levy (Poisson) algebra. In particular, every quantum thermal noise is the orthogonal sum of a quantum Wiener noise and a quantum Poisson noise as it is stated by the Levy-Khinchin theorem in the classical case.
A simple axiomatic characterization of the noncommutative Ito algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. It is proved that every quotient Ito algebra has a faithful representation in a Minkowski space and is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Levy (Poisson) algebra. In particular, every quantum thermal noise of a finite number of degrees of freedom is the orthogonal sum of a quantum Wiener noise and a quantum Poisson noise as it is stated by the Levy-Khinchin theorem in the classical case. Two basic examples of non-commutative Ito finite group algebras are considered.