### 统计代写|随机分析作业代写stochastic analysis代写|Fock Scalar White Noise

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## 统计代写|随机分析作业代写stochastic analysis代写|Fock Scalar White Noise

Definition 1. The standard d-dimensional Fock scalar White Noise $(W N)$ is defined by a quadruple
$$\left{\mathcal{H}, b_{t}, b_{t}^{+}, \Phi\right} ; \quad t \in \mathbb{R}^{d}$$
where $\mathcal{H}$ is a Hilbert space, $\Phi \in \mathcal{H}$ a unit vector called the (Fock) vacusm, and $b_{t}, b_{t}^{+}$are operator valued distributions (for an explanation of this notion see the comment at the end of the present section and the discussion in [AcLuVo02], Section (2.1)) with the following properties.
The vectors of the form
$$b_{t_{n}}^{+} \cdots b_{t_{1}}^{+} \Phi$$
called the number vectors are well defined in the distribution sense and total in $\mathcal{H}$.
$b_{t}$ is the adjoint of $b_{t}^{+}$on the linear span of the number vectors
$$\left(b_{t}^{+}\right)^{+}=b_{t}$$
Weakly on the same domain and in the distribution sense:
$$\left[b_{s}, b_{t}^{+}\right]:=b_{s} b_{t}^{+}-b_{t}^{+} b_{s}=\delta(t-s)$$

where, here and in the following, the symbol $[\cdot, \cdot]$ will denote the commutator:
$$[A, B]:=A B-B A$$
Finally $b_{t}$ and $\Phi$ are related by the Fock property (always meant in the distribution sense):
$$b_{t} \Phi=0$$
The unit vector $\Phi$ determines the expectation value
$$\langle\Phi, X \Phi\rangle=:\langle X\rangle$$
which is well defined for any operator $X$ acting on $\mathcal{H}$ and with $\Phi$ in its domain.
Remark. In the Fock case algebra implies statistics in the sense that the algebraic rules (3.3), (3.2), (3.4) uniquely determine the restriction of the expectation value (3.5) on the polynomial algebra generated by $b_{t}$ and $b_{t}^{+}$. This is because, with the notation
$$X^{\varepsilon}=\left{\begin{array}{l} X, \varepsilon=-1 \ X^{*}, \varepsilon=+1 \end{array}\right.$$
the Fock prescription (3.4) implies that the expectation value
$$\left\langle b_{t_{n}}^{c_{n}} \cdots b_{t_{1}}^{c_{1}}\right\rangle$$
of any monomial in $b_{t}$ and $b_{t}^{+}$is zero whenever either $n$ is odd or $b_{t_{1}}^{c_{1}}=b_{t_{1}}$ or $b_{t_{n}}^{\varepsilon_{n}}=b_{t_{n}}^{+}$. If neither of these conditions is satisfied, then there is a $k \in$ ${2, \ldots, n}$ such that the expectation value $(3.7)$ is equal to
$$\left\langle b_{t_{n}}^{\varepsilon_{n}} \cdots b_{t_{1}}^{\varepsilon_{1}}\right\rangle=\left\langle b_{t_{n}}^{\varepsilon_{n}} \cdots b_{t_{k+1}}^{\varepsilon_{k+1}}\left[b_{t_{k}}, b_{t_{k-1}}^{+} \cdots b_{t_{1}}^{+}\right]\right\rangle$$
Using the derivation property of the commutator $\left[b_{t_{k}}, \cdot\right]$ (i.e. (7.4)) one then reduces the expectation value (3.8) to a linear combination of expectation values of monomials of order less or equal than $n-2$. Iterating one sees that only the scalar term can give a nonzero contribution.

## 统计代写|随机分析作业代写stochastic analysis代写| Classical Real Valued White Noise

Lemma. Let $b_{t}, b_{t}^{+}$be a Fock scalar white noise. Then
$$w_{t}:=b_{t}+b_{t}^{+}$$
is a classical real random variable valued distribution satisfying:
$$\begin{gathered} w_{t}=w_{t}^{+} \ {\left[w_{s}, w_{t}\right]=0 ; \quad \forall s, t} \ \left\langle w_{t}\right\rangle=0 \ \left\langle w_{s} w_{t}\right\rangle=\delta(t-s) \ \left\langle w_{t_{2 n} \ldots} \ldots w_{t_{1}}\right\rangle=\sum_{\left{l_{\alpha}, r_{\alpha}\right} \in p \cdot p-{1, \ldots, 2 n}} \prod_{\alpha=1}^{n}\left\langle w_{t_{l_{\alpha}}} w_{t_{r_{\alpha}}}\right\rangle \end{gathered}$$
moreover all odd moments vanish and $p \cdot p \cdot{1, \ldots, 2 n}$ denotes the set of all pair partitions of ${1, \ldots, 2 n}$.

Remark. The self-adjointness condition (4.2) and the commutativity condition (4.3) mean that $\left(w_{t}\right)$ is (isomorphic to) a classical real valued process. Conditions (4.4) and (4.5) mean respectively that $\left(w_{t}\right)$ is mean zero and $\delta$-correlated. Finally (4.6), which follows from (3.4) and from the same arguments used to deduce the explicit form of (3.7), shows that the classical process $\left(w_{t}\right)$ is Gaussian.

Definition 2. The process $\left(w_{t}\right)$ satisfying (4.2),…, (4.5) (one can prove its uniqueness up to stochastic equivalence) is called the standard $d$-dimensional classical real valued White Noise $(W N)$. The identity (4.1) is called the quantum decomposition of the classical d-dimensional white noise.

Remark. Notice that, for the classical process $\left(w_{t}\right)$, it is not true that algebra implies statistics: this becomes true only using the quantum decomposition (4.1) combined with the Fock prescription (3.4).

Remark. In the case $d=1$, integrating the classical WN one obtains the classical Brownian motion with zero initial condition:
$$W_{t}=B_{t}+B_{t}^{+}=\int_{0}^{t} d s\left(b_{s}^{+}+b_{s}\right)$$
Notice that (4.7) gives the $q$-decomposition of the classical BM just as (4.1) gives the $q$-decomposition of the classical WN.
From now on we will only consider the case $d=1$.

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

An important generalization of the quantum decomposition (4.1) of the classical white noise is the identity:
$$p_{t}(\lambda)=b_{t}+b_{t}^{+}+\lambda b_{t}^{+} b_{t} ; \quad \lambda \geq 0$$
which can be shown to define (in the sense of vacuum distribution) a 1-parameter family of classical real valued distribution processes (i.e. $p_{t}(\lambda)=$ $p_{t}(\lambda)^{+}$and $\left.\left[p_{s}(\lambda), p_{t}(\lambda)\right]=0\right)$. In fact this classical process can be identified, up to a time rescaling, to the compensated scalar valued standard classical Poisson noise with intensity $1 / \lambda$ and the identity (5.1) gives a $q$-decomposition of this process.

Integrating (5.1), in analogy with (4.7), one obtains the standard compensated Poisson processes. Notice that the critical value
$$\lambda=0$$
corresponds to the classical WN while any other value
$$\lambda \neq 0$$
gives a Poisson noise. As a preparation to the discussion of Section (17) notice that $\lambda=0$ is the only critical point, i.e. a point where the vacuum distribution changes and that these two classes of stochastic processes exactly coincide with the first two Meixner classes.

## 统计代写|随机分析作业代写stochastic analysis代写|Fock Scalar White Noise

\left{\mathcal{H}, b_{t}, b_{t}^{+}, \Phi\right} ; \quad t \in \mathbb{R}^{d}\left{\mathcal{H}, b_{t}, b_{t}^{+}, \Phi\right} ; \quad t \in \mathbb{R}^{d}

b吨n+⋯b吨1+披

b吨是的伴随b吨+在数向量的线性跨度上
(b吨+)+=b吨

[bs,b吨+]:=bsb吨+−b吨+bs=d(吨−s)

[一种,乙]:=一种乙−乙一种

b吨披=0

⟨披,X披⟩=:⟨X⟩

$$X^{\varepsilon}=\left{X,e=−1 X∗,e=+1\对。 吨H和F这Cķpr和sCr一世p吨一世这n(3.4)一世米pl一世和s吨H一种吨吨H和和Xp和C吨一种吨一世这n在一种l在和 \left\langle b_{t_{n}}^{c_{n}} \cdots b_{t_{1}}^{c_{1}}\right\rangle 这F一种n是米这n这米一世一种l一世nb吨一种ndb吨+一世s和和r这在H和n和在和r和一世吨H和rn一世s这dd这rb吨1C1=b吨1这rb吨nen=b吨n+.一世Fn和一世吨H和r这F吨H和s和C这nd一世吨一世这ns一世ss一种吨一世sF一世和d,吨H和n吨H和r和一世s一种ķ∈$$2,…,n$s在CH吨H一种吨吨H和和Xp和C吨一种吨一世这n在一种l在和$(3.7)$一世s和q在一种l吨这 \left\langle b_{t_{n}}^{\varepsilon_{n}} \cdots b_{t_{1}}^{\varepsilon_{1}}\right\rangle=\left\langle b_{t_{n }}^{\varepsilon_{n}} \cdots b_{t_{k+1}}^{\varepsilon_{k+1}}\left[b_{t_{k}}, b_{t_{k-1} }^{+} \cdots b_{t_{1}}^{+}\right]\right\rangle$\$

## 统计代写|随机分析作业代写stochastic analysis代写| Classical Real Valued White Noise

\begin{聚集} w_{t}=w_{t}^{+} \ {\left[w_{s}, w_{t}\right]=0 ; \quad \forall s, t} \ \left\langle w_{t}\right\rangle=0 \ \left\langle w_{s} w_{t}\right\rangle=\delta(ts) \ \left\ langle w_{t_{2 n} \ldots} \ldots w_{t_{1}}\right\rangle=\sum_{\left{l_{\alpha}, r_{\alpha}\right} \in p \cdot p-{1, \ldots, 2 n}} \prod_{\alpha=1}^{n}\left\langle w_{t_{l_{\alpha}}} w_{t_{r_{\alpha}}} \right\rangle \end{聚集}\begin{聚集} w_{t}=w_{t}^{+} \ {\left[w_{s}, w_{t}\right]=0 ; \quad \forall s, t} \ \left\langle w_{t}\right\rangle=0 \ \left\langle w_{s} w_{t}\right\rangle=\delta(ts) \ \left\ langle w_{t_{2 n} \ldots} \ldots w_{t_{1}}\right\rangle=\sum_{\left{l_{\alpha}, r_{\alpha}\right} \in p \cdot p-{1, \ldots, 2 n}} \prod_{\alpha=1}^{n}\left\langle w_{t_{l_{\alpha}}} w_{t_{r_{\alpha}}} \right\rangle \end{聚集}

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

p吨(λ)=b吨+b吨++λb吨+b吨;λ≥0

λ=0

λ≠0

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