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# 数学代写|随机过程Stochastic Porcess代考|Wide-sense jump processes

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## 数学代写|随机过程代写Stochastic Porcess代考|Wide-sense jump processes

Wide-sense jump processes. One may expect that in sufficiently regular cases a Markov process with a finite number of states serves as a model of the following process: a moving point is situated at the initial state during certain random intervals of time; after that the point moves according to a given probability law into one of the other possible states; the point stays in this state for a random interval of time and then moves into a new state, and so on.
These types of processes can also be considered in an arbitrary phase space; they are called jump Markov processes.
Consider a wide-sense Markov process in the phase space ${\mathscr{X}, \mathfrak{B}}$ with transition probabilities $P(s, x, t, B), s<t$ and $(s, t) \in \mathscr{I} \times \mathscr{I}$. We shall assume that the $\sigma$-algebra $\mathfrak{B}$ contains all the singletons of $\mathscr{X}$ and we shall consider the noncut-off processes only.
Definition 5. A wide-sense Markov process is called a jump process if for arbitrary $(s, x, B) \in \mathscr{I} \times \mathscr{X} \times \mathfrak{B}$ the limit
$$\lim _{t \downarrow s} \frac{P(s, x, t, B)-\chi(B, x)}{t-s}=\bar{a}(s, x, B)$$
exists and $\bar{a}(s, x, B)$ is a finite charge on $\mathfrak{B}$ for fixed $(s, x)$
A jump wide-sense Markov process is called regular if the convergence in formula $(23)$ is uniform in $(s, x, B) \in[0, t] \times \mathscr{X} \times \mathfrak{B}$ and the function $\bar{a}(s, x, B)$ for fixed $(x, B)$ is continuous in $s \in[0, t]$ uniformly in $(x, B)$ where $t$ is an arbitrary point in $\mathscr{I}$.
We note that the function $\bar{a}(s, x, B)$ satisfies
$$\begin{gathered} \bar{a}(s, x, \mathscr{X})=0, \quad \bar{a}(s, x, B)=\lim {t \downarrow s} \frac{P(s, x, t, B)}{t-s} \geqslant 0, \quad \text { if } \quad x \notin B, \ \bar{a}(s, x,{x})=-\bar{a}(s, x, \mathscr{X} \backslash{x})=\lim {t \downarrow s} \frac{P(s, x, t,{x})-1}{t-s} \leqslant 0, \end{gathered}$$
where ${x}$ is the singleton containing $x$. These relations can be combined into a single formula by setting
where
$$\bar{a}(s, x, B)=-a(s, x) \chi(B, x)+a(s, x, B)$$
$$a(s, x)=-\bar{a}(s, x,{x}), \quad a(s, x, B)=\bar{a}(s, x, B \backslash{x})$$
moreover, $a(s, x, B)$ is a finite measure on $\mathfrak{B}$ and $a(s, x,{x})=0$.

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

Transition probabilities $P(s, x, t, B)$ of a regular jump process are differentiable with respect to $s, s<t$, and satisfy the equation
$$\frac{\partial P(s, x, t, B)}{\partial s}=a(s, x)\left[P(s, x, t, B)-\int P(s, y, t, B) \Pi(s, x, d y)\right]$$
with the boundary condition
$$\lim _{s \uparrow t} P(s, x, t, B)=\chi(B, x)$$
We now show that under certain conditions the functions $a(t, x)$ and $a(t, x, B)$ uniquely determine a regular jump wide-sense Markov process. We discuss the solutions of equations (29) or (32) under suitable boundary conditions.
Let $\mathscr{I}=\left[0, t^*\right]$. In view of the regularity assumption on the jump process, we impose the following conditions on the functions $a(t, x)$ and $a(t, x, B)$ :
a) for fixed $(t, x) \in \mathscr{I} \times \mathscr{X}$ the function $a(t, x, B)$ is a measure on $\mathfrak{B}, a(t, x,{x})=0$ and $a(t, x)=a(t, x, \mathscr{X})$;
b) for fixed $(x, B)$ the function $a(t, x, B), t \in \mathscr{I}$, is continuous in $t$ uniformly in $(x, B)$ and, for fixed $(t, B)$, it is $\mathfrak{B}$-measurable as a function of $x$.
Denote by $W=W(\mathfrak{B})$ the space of all finite completely additive functions (finite charges) $w(B)$, defined on a measurable space ${\mathscr{X}, \mathfrak{B}}$. Define a distance on $W$ by means of the relation
$$\varrho\left(w_1, w_2\right)=\left|w_1(B)-w_2(B)\right|, \quad w_i \in W,$$
where
$$|w(B) ## 随机过程代写 ## 数学代写|随机过程代写Stochastic Porcess代考|Wide-sense jump processes 广义跳跃过程。人们可以期望，在足够规则的情况下，具有有限状态数的马尔可夫过程可以作为以下过程的模型:在一定的随机时间间隔内，一个移动点位于初始状态;之后，该点根据给定的概率定律移动到其他可能的状态之一;点在这个状态中保持一段随机的时间间隔，然后进入一个新的状态，以此类推。 这些类型的过程也可以在任意相空间中考虑;它们被称为跳跃马尔可夫过程。 考虑一个广义马尔可夫过程在相空间{\mathscr{X}, \mathfrak{B}}与跃迁概率P(s, x, t, B), s<t和(s, t) \in \mathscr{I} \times \mathscr{I}。我们将假设\sigma -代数\mathfrak{B}包含\mathscr{X}的所有单例，我们将只考虑非截止过程。 定义:广义马尔可夫过程称为跳跃过程，如果对于任意(s, x, B) \in \mathscr{I} \times \mathscr{X} \times \mathfrak{B}极限$$
\lim _{t \downarrow s} \frac{P(s, x, t, B)-\chi(B, x)}{t-s}=\bar{a}(s, x, B)
$$对于固定的(s, x)，存在并且\bar{a}(s, x, B)是\mathfrak{B}上的有限电荷 如果公式(23)中的收敛性在(s, x, B) \in[0, t] \times \mathscr{X} \times \mathfrak{B}中是一致的，并且固定(x, B)的函数\bar{a}(s, x, B)在s \in[0, t]中连续在(x, B)中是一致的，则跳跃广义马尔可夫过程称为正则过程，其中t是\mathscr{I}中的任意点。 我们注意到函数\bar{a}(s, x, B)满足$$
\begin{gathered}
\bar{a}(s, x, \mathscr{X})=0, \quad \bar{a}(s, x, B)=\lim {t \downarrow s} \frac{P(s, x, t, B)}{t-s} \geqslant 0, \quad \text { if } \quad x \notin B, \ \bar{a}(s, x,{x})=-\bar{a}(s, x, \mathscr{X} \backslash{x})=\lim {t \downarrow s} \frac{P(s, x, t,{x})-1}{t-s} \leqslant 0,
\end{gathered}
$$其中{x}是包含x的单例。这些关系可以通过设置组合成一个公式 在哪里$$
\bar{a}(s, x, B)=-a(s, x) \chi(B, x)+a(s, x, B)

a(s, x)=-\bar{a}(s, x,{x}), \quad a(s, x, B)=\bar{a}(s, x, B \backslash{x})


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


\压裂{\部分P (x, t, B)}{\部分年代}=左(年代,x) \ [P (s, x, t, B) – \ int P (s, y, t, B) \π(x, y) ]



P(s, x, t, B)=\chi(B, x)


a)对于固定$(t, x) \in \mathscr{I} \times \mathscr{x}$，函数$a(t, x, B)$是$\mathfrak{B}， a(t, x，{x})=0$和$a(t, x)=a(t, x， \mathscr{x})$的测度;
b)对于固定$(x, b)$，函数$a(t, x, b)， t \in \mathscr{I}$，在$t$中连续一致地在$(x, b)$中连续，对于固定$(t, b)$，它是$\mathfrak{b}$-可测量的$x$的函数。


\ varrho \离开(w_1 w_2 \右)= \左| w_1 (B) -w_2 (B) \ |, \四w_i \ W,



| w (B)

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