### 统计代写|随机分析作业代写stochastic analysis代写|Itô Calculus and Quantum White Noise Calculus

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## 统计代写|随机分析作业代写stochastic analysis代写|Itô Calculus and Quantum White Noise Calculus

Luigi Accardi ${ }^{1}$ and Andreas Boukas ${ }^{2}$
1 Centro Vito Volterra, Università di Roma Tor Vergata via Columbia, 2-00133
Roma, Italy. accardievolterra,mat. uniroma2 it,
http: //volterra mat . uniroma2. it
2 Department of Mathematics and Natural Sciences, American College of Greece, Aghia Paraskevi, Athens 15342, Greece, andreasboukaslacgmail.gr

Summary. Itô calculus has been generalized in white noise analysis and in quantum stochastic calculus. Quantum white noise calculus is a third generalization, unifying the two above mentioned ones and bringing some unexpected insight into some old problems studied in different fields, such as the renormalization problem in physics and the representation theory of Lie algebras. The present paper is an attempt to explain the motivations of these extensions with emphasis on open challenges.
The last section includes a result obtained after the Abel Symposium. Namely that, after introducing a new renormalization technique, the RHPWN Lie algebra includes (in fact we will prove elsewhere that this inclusion is an identification) a second quantized version of the extended Virasoro algebra, i.e. the VirasoroZamolodchikov *-Lie algebra $w_{\infty}$, which has been widely studied in string theory and in conformal field theory.

## 统计代写|随机分析作业代写stochastic analysis代写|

Luigi Accardi $^{1}$ and Andreas Boukas ${ }^{2}$
1 Centro Vito Volterra, Università di Roma Tor Vergata via Columbia, 2-00133 Roma, Italy. accardiavolterra .mat uniroma2. it, http://volterra.mat . uniroma2. it
2 Department of Mathematics and Natural Sciences, American College of Greece, Aghia Paraskevi, Athens 15342, Greece, andreasboukaslacgmail . gr

Summary. Itô calculus has been generalized in white noise analysis and in quantum stochastic calculus. Quantum white noise calculus is a third generalization, unifying the two above mentioned ones and bringing some unexpected insight into some old probloms studied in different fields, such as the renormalization problem in physics and the representation theory of Lie algebras. The present paper is an attempt to explain the motivations of these extensions with emphasis on open challenges.
The last section includes a result obtained after the Abel Symposium. Namely that, after introducing a new renormalization technique, the RHPWN Lie algebra includos (in foct we will prove elsewhere that this incluzion is an identifiontion) a second quantized version of the extended Virasoro algebra, i.e. the VirasoroZamolodchikov *-Lie algebra $w_{\infty}$, which has been widely studied in string theory and in conformal field theory.

## 统计代写|随机分析作业代写stochastic analysis代写|Plan of the Present Paper

The goal of the present section is twofold: (i) to give a more analytical outline of the content of the present paper; (ii) to catch this occasion to say a few words about the motivations and the inner logic underlying the developments described here as well as about their connections with other sectors of quantum probability which could not be dealt with for reasons of space.
Section (3) defines the notion of quantum (Boson Fock) white noise and illustrates, in this basic particular case, one of the main ideas of quantum probability, i.e. the idea that algebra implies statistics. Let me just mention here that also the converse statement, i.e. that statistics implies algebra (e.g. commutation or anti commutation relations), is true and it lies at a deeper level. The first result in this direction was proved by von Waldenfels in the Bose and Fermi case [voWaGi78, voWa78] and about 20 years later, with the introduction of the notion of interacting Fock space [AcLuVo97b], this principle became a quite universal principle of probability theory and opened the way to the program of a full algebraic classification of probability measures. This is a quite interesting direction, and is also deeply related to the main topic of the present paper, stochastic and white noise calculus, but we will not discuss this connection and we refer the interested reader to $[\mathrm{AcB} \mathrm{B} 98$, AcKuSt02, AcKuSt05a].

Section (4) describes another important new idea of quantum probability, i.e. the notion of quantum decomposition of a classical random variable (or stochastic process). This idea is illustrated in the important particular case of classical white noise and extended, in Section (6), to the Poisson noise.

The two above mentioned decompositions are at the root of HudsonParthasarathy’s quantum extension of classical Itô calculus, briefly outlined in Section (6).

Section (7) briefly describes the classical Schrödinger and Heisenberg equations as a preparation to their stochastic and white noise versions.

The algebraic form of a classical stochastic process is described in Section (8). This leads to a reformulation, explained in Section (10), of classical stochastic differential equations, that makes quite transparent their equivalence to stochastic versions of the classical Schrödinger or of Heisenberg equations.

## 统计代写|随机分析作业代写stochastic analysis代写|Itô Calculus and Quantum White Noise Calculus

1 Vito Volterra Center, University of Rome Tor Vergata via Columbia, 2 – 00133
Rome, Italy。艾卡迪沃尔泰拉垫子 uniroma2 它，
http://volterra mat。uniroma2. it

## 统计代写|随机分析作业代写stochastic analysis代写|

1 Vito Volterra Center, University of Rome Tor Vergata via Columbia, 2 – 00133 Rome, Italy。accardiavolterra .mat uniroma2。它，http://volterra.mat。uniroma2.

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