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# 数学代写|随机过程Stochastic Porcess代考|Markov Processes

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## 数学代写|随机过程代写Stochastic Porcess代考|Markov Processes

Definition. The definition of a Markov function given above is inadequate in many cases. Firstly we are often required to introduce a probabilistic object which is a collection of interconnected random processes. This happens, for example, when a movement (evolution) of a certain system is considered in its phase space; this movement may start at an arbitrary instant of time from an arbitrary point of the phase space and one is required to study the totality of possible movements.

The notion of a Markov process is introduced for investigation of such objects. Generally speaking, its description is as follows: we define a predetermined function $\xi(t, \omega)$ of time $t$ and elementary events $\omega$ with values in the phase space of the system and a collection of probability measures $\mathrm{P}_{s, x}$ each one of which determines probabilistic properties of a movement starting at time $s$ from the point $x$. Such a description is feasible in view of the Markov property of the process (the absence of the aftereffect): if at the instant of time $s$, the system was situated at point $x$, then its further evolution is completely determined by the measure $\mathrm{P}_{s, x}$ and is independent of the additional information about the movement of the system up to the instant $s$.

Next it is reasonable when defining a Markov process to single out explicitly the special state of the system which corresponds to the departure of the system from the phase space (“into infinity”, or “death” of the system).

As far as the time interval $\mathscr{I}$ is concerned it will always be assumed in the remainder of this Chapter (except for the last article of the present section) that $\mathscr{I}=[0, \infty)$.
Thus we are given:
a) a measurable space ${\mathscr{X}, \mathfrak{B}}$ and a point $\mathfrak{v} \notin \mathscr{X}$; the space ${\mathscr{X}, \mathfrak{B}}$ (or $\mathscr{X}$ ) is called the phase space of the system (process); we set $\mathscr{X}{\mathfrak{v}}=\mathscr{X} \cup{\mathfrak{v}}$ and denote by $\mathfrak{B}{\mathfrak{v}}$ the minimal $\sigma$-algebra in $\mathscr{X}{\mathfrak{v}}$ containing $\mathfrak{B}$ and the singleton containing the point $v$. b) a measurable space ${\Omega, \widetilde{\Im}}$ and a family of $\sigma$-algebras $\left{\widetilde{\Xi}_t^s, 0 \leqslant s \leqslant t \leqslant \infty\right}$, place of $\widetilde{S}_t^0$ and $\widetilde{S}^s$ in place of $\widetilde{S}{\infty}^s$;
c) a probability measure $P_{s, x}$ for each pair $(s, x) \in[0, \infty) \times \mathscr{X}{\mathrm{v}}$ on $\mathrm{s}^s$. d) a function $\xi(t, \omega)$ defined on $[0, \infty) \times \Omega$ with values in $\mathscr{X}{\mathrm{v}}$ possessing the following properties: if for some $t_0, \omega_0, \xi\left(t_0, \omega_0\right)=\mathfrak{v}$, then $\xi\left(t, \omega_0\right)=\mathfrak{v}$ also for $t>t_0$.

## 数学代写|随机过程代写Stochastic Porcess代考|Completion of basic O”-algebras

Completion of basic $\sigma$-algebras. It is assumed in the definition of a Markov process that the function $\xi_t(\omega) \in \mathfrak{G}t^s \mid \mathfrak{B}{\mathfrak{v}}(s \leqslant t)$. In certain cases it is important that $\xi_t(\omega)$ be a measurable mapping into the space $\left{\mathscr{X}, \mathfrak{B}_{\mathfrak{v}}^{\prime}\right}$ where $\mathfrak{B}^{\prime}$ is a larger $\sigma$-algebra than $\mathfrak{B}$, or that the relations which are valid for the sets in $\mathfrak{N}t^s$ will also hold for a somewhat wider class of sets. It will be shown in the present section that one can extend $\sigma$-algebras $\widetilde{S}_t^s$ and $\mathfrak{N}_t^s$ in an appropriate manner, retaining their role played in the definition and properties of a Markov process; in particular, $\xi_t(\omega)$ becomes a measurable mapping into $\left{\mathscr{X}{\mathfrak{v}}, \mathfrak{B}_{\mathfrak{v}}^\right}$, where $\mathfrak{B}^$ is in general a substantial extension of the $\sigma$-algebra $\mathfrak{B}$.

We recall the structure of the completion operation of a $\sigma$-algebra with respect to a certain measure. Let $\mathfrak{F}$ be a $\sigma$-algebra and $q$ be a measure on $\mathfrak{F} ; \mathfrak{F}$ is called a $q$-complete provided $A \subset B, B \in \mathfrak{F}$ and $q(B)=0$ implies that $A \in \mathfrak{F}$.
If the $\sigma$-algebra $\mathfrak{F}$ is not complete it can be completed using the following procedure. Define a class of sets $\mathfrak{F}^q$ by setting $A \in \mathfrak{F}^q$ provided $F_1$ and $F_2$ are found belonging to $\mathscr{F}$ such that $F_1 \subset A \subset F_2$ and $q\left(F_2 \backslash F_1\right)=0$.

It is easy to show that the class of sets $\mathfrak{F}^q$ is a $q$-complete $\sigma$-algebra. The sets belonging to $\mathfrak{F}^q$ can be characterized as follows: $A \in \mathfrak{F}^q$ if and only if there exists a set $B \in \mathfrak{F}$ such that $A \triangle B$ is a subset of a certain set in $\mathfrak{F}$ of $q$-measure 0 .
Let $\mathscr{Q}$ be a family of measures. Set
$$\mathfrak{F}^2=\bigcap_{q \in \mathscr{2}} \mathfrak{F}^q$$
and call $\mathfrak{F}^2$ the completion of the $\sigma$-algebra $\mathfrak{F}$ with respect to the family of measures $\mathscr{2}$. If $\mathfrak{F}^2=\mathfrak{F}$ we say that the $\sigma$-algebra $\mathfrak{F}$ is $\mathscr{2}$-complete.

The definition of $\mathfrak{F}^2$ can also be formulated in the following manner: $F \in \mathfrak{F}^2$ if and only if for any measure $q \in \mathscr{2}$ there exists a set $F_q \in \mathfrak{F}$ such that $F \triangle F_q \in \mathfrak{F}^q$ and $q\left(F \triangle F_q\right)=0$.

## MATLAB代写

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