Posted on Categories:Stochastic Porcesses, 数学代写, 随机过程

# 数学代写|随机过程Stochastic Porcess代考|Non-cut-off processes

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## 数学代写|随机过程代写Stochastic Porcess代考|Non-cut-off processes

Let $x(t)$ be a non-cut-off homogeneous locally regular Markov process on $\mathscr{J}$. We shall assume that the process is strong Markov and continuous from the right. Note that if $x(t)$ is separable and strong Markov then it is also continuous from the right (under the condition of local regularity). This follows from the facts that 1) a separable process spends a positive amount of time in each state and 2) for each Markov moment $\tau$ the functions $x(\tau)$ and $x(\tau+s)=x(\tau)$ are defined provided $s<\varepsilon$ for some $\varepsilon>0$. We introduce a transfinite sequence of moments $\xi_\alpha$ which are the transition times of the process from one state to another in the following manner: if $\alpha$ is an ordinal of the first order type, then
$$\xi_\alpha=\xi_{\alpha-1}+\theta_{\xi_{\alpha-1}} \xi_1$$
where $\xi_1$ is the moment of the first exit from the initial state; if, however $\alpha$ is an ordinal of the second order type then
$$\xi_\alpha=\sup {\beta<\alpha} \xi\beta$$
The process $x(t)$ is continuous (constant) on each interval $\left[\xi_\alpha, \xi_{\alpha+1}\right)$ and is therefore continuous from the right: if $t_n \downarrow t_0$ and $t_0 \in\left[\xi_\alpha, \xi_{\alpha+1}\right)$ then starting from some $n, x\left(t_n\right)=x\left(t_0\right)$.

We introduce a transfinite chain of Markov moments in the following manner:
$$\zeta^0=\sum_1^{\infty} \tau_k$$
(the variables $\tau_k$ were introduced above); if $\alpha$ is an ordinal of the first order type then
$$\zeta^\alpha=\sup n \zeta_n^{\alpha-1}, \quad \zeta_1^{\alpha-1}=\zeta^{\alpha-1}, \quad \zeta_n^{\alpha-1}=\zeta^{\alpha-1}+\theta{\zeta^{\alpha-1}} \zeta_{n-1}^{\alpha-1} \quad(n>1)$$
if, however $\alpha$ is an ordinal of the second order type then $\zeta^\alpha=\sup {\beta<\alpha} \zeta^\beta$. Set, for $\lambda>0$, $$\Gamma\lambda^\alpha f(k)=\mathrm{E}_k e^{-\lambda \zeta^\alpha} f\left(x\left(\zeta^\alpha\right)\right)$$

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

Constructive definition of a semi-Markov process. Let $\mathscr{X}$ be an arbitrary space, $\mathfrak{B}$ a $\sigma$-algebra of its subsets which contains all the singletons. Assume that a strong Markov process $\left{\mathscr{F}, \mathcal{N}, \mathrm{P}_x\right}$ is defined on ${\mathscr{X}, \mathfrak{B}}$, with sample function $x(t)$ continuous from the right with respect to the discrete topology on $\mathscr{X}$ induced by the metric
$$r(x, y)= \begin{cases}0 ; & x=y, \ 1 ; & x \neq y .\end{cases}$$
The process spends a positive amount of time in each state and $\tau$, the moment of the first exit from the initial state, is exponentially distributed with parameter $\lambda(x)$ depending, naturally, on the initial state.
Denote
$$\pi(x, B)=\mathrm{P}x{x(\tau) \in B}, \quad B \in \mathfrak{B}$$ The characteristic operator for the process $\left{\mathscr{F}, \mathscr{N}, \mathrm{P}_x\right}$ is defined on all bounded $\mathfrak{B}$-measurable functions $f(x)$ by the relation $$\mathfrak{A} f(x)=\frac{\mathrm{E}_x f\left(x\tau\right)-f(x)}{\mathrm{E}_x \tau}=\lambda(x)\left(\int f(y) \pi(x, d y)-f(x)\right) .$$

## 数学代写|随机过程代写Stochastic Porcess代考|Non-cut-off processes

$$\xi_\alpha=\xi_{\alpha-1}+\theta_{\xi_{\alpha-1}} \xi_1$$

$$\xi_\alpha=\sup {\beta<\alpha} \xi\beta$$

$$\zeta^0=\sum_1^{\infty} \tau_k$$
(上文已经介绍了变量$\tau_k$);如果$\alpha$是一阶类型的序数，则
$$\zeta^\alpha=\sup n \zeta_n^{\alpha-1}, \quad \zeta_1^{\alpha-1}=\zeta^{\alpha-1}, \quad \zeta_n^{\alpha-1}=\zeta^{\alpha-1}+\theta{\zeta^{\alpha-1}} \zeta_{n-1}^{\alpha-1} \quad(n>1)$$

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

$$r(x, y)= \begin{cases}0 ; & x=y, \ 1 ; & x \neq y .\end{cases}$$

$$\pi(x, B)=\mathrm{P}x{x(\tau) \in B}, \quad B \in \mathfrak{B}$$过程$\left{\mathscr{F}, \mathscr{N}, \mathrm{P}_x\right}$的特征算子由关系定义在所有有界的$\mathfrak{B}$ -可测量函数$f(x)$上 $$\mathfrak{A} f(x)=\frac{\mathrm{E}_x f\left(x\tau\right)-f(x)}{\mathrm{E}_x \tau}=\lambda(x)\left(\int f(y) \pi(x, d y)-f(x)\right) .$$

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