### 统计代写|随机分析作业代写stochastic analysis代写|The Hudson-Parthasarathy Quantum Stochastic Calculus

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## 统计代写|随机分析作业代写stochastic analysis代写|The Hudson-Parthasarathy Quantum Stochastic Calculus

In the previous sections we have seen that, integrating the densities
\begin{aligned} w_{t} &=b_{t}+b_{t}^{+} \ p(\lambda){t} &=b{t}+b_{t}^{+}+\lambda b_{t}^{+} b_{t} \end{aligned}
one obtains the stochastic differentials (random measures) as WN integrals
$$\begin{gathered} d W_{t}=\int_{t}^{t+d t} w_{s} d s=\int_{t}^{t+d t}\left(b_{s}+b_{s}^{+}\right) d s=: d B_{t}^{+}+d B_{t} \ d P_{t}(\lambda)=\int_{t}^{t+d t} p_{s}(\lambda) d s=\int_{t}^{t+d t}\left(b_{s}+b_{s}^{+}+\lambda b_{s}^{+} b_{s}\right) d s=d B_{t}^{+}+d B_{t}+\lambda d N_{t} \end{gathered}$$

Starting from these one defines the classical stochastic integrals with the usual constructions.
$$\int_{0}^{t} F_{s} d W_{s} ; \quad \int_{0}^{t} F_{s} d P_{s}(\lambda)$$
The passage to $q$-stochastic integrals consists in separating the stochastic integrals corresponding to the different pieces. In other words, the quantum decomposition (5.1) suggests to introduce separately the stochastic integrals
$$\int_{0}^{t} F_{s} d B_{s} ; \quad \int_{0}^{t} F_{s} d B_{s}^{+} ; \quad \int_{0}^{t} F_{s} d N_{s}$$
This important development was due to Hudson and Parthasarathy and we refer to the monograph [Partha92] for an exposition of the whole theory.

## 统计代写|随机分析作业代写stochastic analysis代写|Schr¨odinger and Heisenberg Equations

A Schrödinger equation (also called an operator Hamiltonian equation) is an equation of the form:
$$\partial_{t} U_{t}=-i H_{t} U_{t} ; \quad U_{0}=1 ; \quad t \in \mathbb{R}$$
where the 1 -parameter family of symmetric operators on a Hilbert space $\mathcal{H}$
$$H_{t}=H_{t}^{*}$$
is called the Hamiltonian. In the pyhsics literature one often requires the positivity of $H_{t}$. We do not follow this convenction in order to give a unified treatment of the usual Schrödinger equation and of its so-called interaction representation form. This approach is essential to underline the analogy with the white noise Hamiltonian equations, to be discussed in Section (12).

When $H_{t}$ is a self-adjoint operator independent of $t$, the solution of equation (7.1) exists and is a 1 -parameter group of unitary operators:
$$U_{t} \in U n(\mathcal{H}) ; U_{s} U_{t}=U_{s+t} ; U_{0}=1 ; U_{t}^{*}=U_{t}^{-1}=U_{-t} ; s, t \in \mathbb{R}$$
Conversely every 1-parameter group of unitary operators is the solution of equation (7.1) for some self-adjoint operator $H_{t}=H$ independent of $t$.
An Heisenberg equation, associated to equation (7.1), is
$$\partial_{t} X_{t}=\delta_{t}\left(X_{t}\right) ; \quad X_{0}=X \in \mathcal{B}(\mathcal{H})$$
where $\delta_{t}$ has the form
$$\delta_{t}\left(X_{t}\right):=-i\left[H_{t}, X_{t}\right] ; \quad X_{0}=X \in \mathcal{B}(\mathcal{H})$$
One can prove that $\delta_{t}$ is a *-derivation, i.e. a linear operator on an appropriate subspace of the algebra $\mathcal{B}(\mathcal{H})$ of all the bounded operators on $\mathcal{H}$, also called the algebra of observables, satisfying (on this subspace):

$$\begin{gathered} \delta_{t}(a b)=\delta_{t}(a) b+a \delta_{t}(b) \ \delta_{t}^{}(a):=\delta_{t}\left(a^{}\right)^{*}=\delta_{t}(a) \end{gathered}$$
Not all *-derivations $\delta_{t}$ on subspaces (or sub algebras) of $\mathcal{B}(\mathcal{H}$ ) have the form (7.3). If this happens, then the $-$ derivation, $\delta_{t}$, and sometimes also the Heisenberg equation, is called inner and its solution has the form $$X_{t}=U_{t} X_{t} U_{t}^{}$$
where $U_{t}$ is the solution of the corresponding Schrödinger equation (7.1). Conversely, every solution $U_{t}$ of the Schrödinger equation (7.1) defines, through (7.5), a solution of the Heisenberg equation (7.2) with $\delta_{t}$ given by (7.3).

Thus every Schrödinger equation is canonically associated to an Heisenberg equation. The converse is in general false, i.e. there are Heisenberg equations with no associated Schrödinger equation (equivalently: not always a derivation is inner). The simplest physically relevant examples of this situation are given by the quantum generalization of the so called interacting particle systems [AcKo00b] which have been widely studied in classical probability.

## 统计代写|随机分析作业代写stochastic analysis代写|Algebraic Form of a Classical Stochastic Process

Let $\left(X_{t}\right)$ be a real valued stochastic process. Define
$$j_{t}(f):=f\left(X_{t}\right)$$
In the spirit of quantum probability, we realize $f$ as a multiplication operator on $L^{2}(\mathbb{R})$ and $f\left(X_{t}\right)$ as a multiplieation operator on
$$L^{2}\left(\mathbb{R} \times \Omega, \mathcal{B}{\mathbb{Z}} \times \mathcal{F}{+} d x \otimes P\right) \equiv L^{2}(\mathbb{R}) \otimes L^{2}\left(\Omega, \mathcal{F}{,} P\right)$$ where $(\Omega, \mathcal{F}, P)$ is the probability space of the process $\left(X{t}\right)$ and $\mathcal{B}{\mathbb{R}}$ denotes the Borel $\sigma$-algebra on RR. Sometimes we use the notation: $$M{f} \varphi(x):=f(x) \varphi(x) ; \quad \varphi \in L^{2}(\mathbb{R})$$
The same notation will be used if $x \in \mathbb{R}$ is replaced by $(x, \omega) \in \mathbb{R} \times \Omega$.
Thus $f\left(X_{t}\right)$ is realized as multiplication operator on $L^{2}(\mathbb{R}) \otimes L^{2}(\Omega, \mathcal{F}, P)$. With these notations, for each $t \geq 0, j_{t}$ is a $*$-homomorphism
$$j_{t}: \mathcal{C}^{2}(\mathbb{R}) \subseteq \mathcal{B}\left(L^{2}(\mathbb{R})\right) \rightarrow \mathcal{B}\left(L^{2}(\mathbb{R}) \otimes L^{2}(\Omega, \mathcal{F}, P)\right)$$

## 统计代写|随机分析作业代写stochastic analysis代写|The Hudson-Parthasarathy Quantum Stochastic Calculus

\begin{aligned} w_{t} &=b_{t}+b_{t}^{+} \ p(\lambda) {t} &=b {t}+b_{t}^{+}+\lambda b_{t}^{+} b_{t} \end{对齐} 这n和这b吨一种一世ns吨H和s吨这CH一种s吨一世Cd一世FF和r和n吨一世一种ls(r一种nd这米米和一种s在r和s)一种s在ñ一世n吨和Gr一种ls d在吨=∫吨吨+d吨在sds=∫吨吨+d吨(bs+bs+)ds=:d乙吨++d乙吨 d磷吨(λ)=∫吨吨+d吨ps(λ)ds=∫吨吨+d吨(bs+bs++λbs+bs)ds=d乙吨++d乙吨+λdñ吨

∫0吨Fsd在s;∫0吨Fsd磷s(λ)

∫0吨Fsd乙s;∫0吨Fsd乙s+;∫0吨Fsdñs

## 统计代写|随机分析作业代写stochastic analysis代写|Schr¨odinger and Heisenberg Equations

∂吨在吨=−一世H吨在吨;在0=1;吨∈R

H吨=H吨∗

∂吨X吨=d吨(X吨);X0=X∈乙(H)

d吨(X吨):=−一世[H吨,X吨];X0=X∈乙(H)

## 统计代写|随机分析作业代写stochastic analysis代写|Algebraic Form of a Classical Stochastic Process

j吨(F):=F(X吨)

j吨:C2(R)⊆乙(大号2(R))→乙(大号2(R)⊗大号2(Ω,F,磷))

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