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数学代写|随机分析代写Stochastic Calculus代考|MATH4512 UN CIG JI and NOBUAKI OBATA

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数学代写|随机分析代写Stochastic Calculus代考|UN CIG JI and NOBUAKI OBATA

The so-called “white noise calculus” is a well-known area of stochastic analysis. It was the basic motivation of Hida [30] to construct a suitable framework where the white noise, i.e., the time derivative of Brownian motion, is realized as a stochastic process with explicit time parameter. Since then a large number of works related to white noise calculus have been published covering many topics with various backgrounds, see Kuo [66] for the collection of references up to 2002 . The diversity of white noise calculus has been accelerated since it encountered quantum probability; A quantum aspect was explicitly introduced into white noise calculus and a white noise approach to quantum stochastic analysis was launched out by Obata [77-79]. Then “quantum white noise calculus” has developed along with normal-ordered white noise differential equations as an extension of quantum stochastic differential equations of Itô type $[17-19,79,80]$, quantum stochastic integrals of Itô type and quantum Itô formula $[36,37]$, their extensions to higher powers of quantum white noise $[16]$, integral representation of quantum martingales [41], quantum stochastic gradients and quantum Hitsuda-Skorohod integrals [51], quantum white noise derivatives $[46,48]$ with their applications to quantum martingales [49] and to the implementation problem of the canonical commutation relations [52]. The basis of these developments is found in the operator theory on white noise functions [76], established in the early 1990’s. There is a survey on quantum white noise calculus by the same authors [42], which complements the results obtained until 2002 . The main purpose of this paper is to provide a concise access to “quantum white noise calculus” and to show some of the recent achievements.

数学代写|随机分析代写Stochastic Calculus代考|Standard construction of countable Hilbert spaces

Let $H_{\mathbb{R}}$ be a real Hilbert space with an inner product $\langle\cdot, \cdot\rangle$, and $H=$ $H_{\mathbb{R}}+i H_{\mathbb{R}}$ the complexification. Since the inner product of $H_{\mathbb{R}}$ is an $\mathbb{R}$ bilinear form on $H_{\mathbb{R}} \times H_{\mathbb{R}}$, it is naturally extended to a $\mathbb{C}$-bilinear form on $H \times H$, denoted by the same symbol. Then the inner product of $H$ is defined by
$$\langle\xi \mid \eta\rangle=\langle\bar{\xi}, \eta\rangle, \quad \xi, \eta \in H .$$
Obviously, $\langle\xi \mid \eta\rangle=\langle\xi, \eta\rangle$ for $\xi, \eta \in H_{\mathbb{R}}$. The norm of $H$ is defined by
$$|\xi|0=\sqrt{\langle\xi \mid \xi\rangle}, \quad \xi \in H .$$ The norm of $H{\mathbb{R}}$ is defined in a similar manner and is denoted by the same symbol.

A chain of Hilbert spaces rigging $H$ is constructed by means of a positive selfadjoint operator in a standard manner. Let $A$ be a selfadjoint operator with dense domain $\operatorname{Dom}(A) \subset H$, which is positive, i.e., $\inf \operatorname{Spec}(A)>0$, and real, i.e., $A$ maps $\operatorname{Dom}(A) \cap H_{\mathbb{R}}$ into $H_{\mathbb{R}}$. For each $p \geq 0$, the dense subspace Dom $\left(A^p\right) \subset H$ becomes a Hilbert space equipped with the norm
$$|\xi|_p=\left|A^p \xi\right|_0, \quad \xi \in \operatorname{Dom}\left(A^p\right)$$

数学代写|随机分析代写Stochastic Calculus代考|Standard construction of countable Hilbert spaces

$$\langle\xi \mid \eta\rangle=\langle\bar{\xi}, \eta\rangle, \quad \xi, \eta \in H .$$

$$|\xi| 0=\sqrt{\langle\xi \mid \xi\rangle}, \quad \xi \in H .$$

$$|\xi|_p=\left|A^p \xi\right|_0, \quad \xi \in \operatorname{Dom}\left(A^p\right)$$

MATLAB代写

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