### 统计代写|蒙特卡洛方法代写monte carlo method代考| Markov Jump Processes

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• Foundations of Data Science 数据科学基础

## 统计代写|蒙特卡洛方法代写monte carlo method代考|Markov Jump Processes

A Markov jump process $X=\left{X_{t}, t \geqslant 0\right}$ can be viewed as a continuous-time generalization of a Markov chain and also of a Poisson process. The Markov property $(1.30)$ now reads
$$\mathbb{P}\left(X_{t+s}=x_{t+s} \mid X_{u}=x_{u}, u \leqslant t\right)=\mathbb{P}\left(X_{t+s}=x_{t+s} \mid X_{t}=x_{t}\right) .$$
As in the Markov chain case, one usually assumes that the process is timehomogeneous, that is, $\mathbb{P}\left(X_{t+s}=j \mid X_{t}=i\right)$ does not depend on $t$. Denote this probability by $P_{s}(i, j)$. An important quantity is the transition rate $q_{i j}$ from state $i$ to $j$, defined for $i \neq j$ as
$$q_{i j}=\lim {t \downarrow 0} \frac{P{t}(i, j)}{t} .$$
The sum of the rates out of state $i$ is denoted by $q_{i}$. A typical sample path of $X$ is shown in Figure 1.6. The process jumps at times $T_{1}, T_{2}, \ldots$ to states $Y_{1}, Y_{2}, \ldots$, staying some length of time in each state.

More precisely, a Markov jump process $X$ behaves (under suitable regularity conditions; see [3]) as follows:

1. Given its past, the probability that $X$ jumps from its current state $i$ to state $j$ is $K_{i j}=q_{i j} / q_{i}$.
2. The amount of time that $X$ spends in state $j$ has an exponential distribution with mean $1 / q_{j}$, independent of its past history.

The first statement implies that the process $\left{Y_{n}\right}$ is in fact a Markov chain, with transition matrix $K=\left(K_{i j}\right)$.

A convenient way to describe a Markov jump process is through its transition rate graph. This is similar to a transition graph for Markov chains. The states are represented by the nodes of the graph, and a transition rate from state $i$ to $j$ is indicated by an arrow from $i$ to $j$ with weight $q_{i j}$.

## 统计代写|蒙特卡洛方法代写monte carlo method代考|Birth-and-Death Process

A birth-and-death process is a Markov jump process with a transition rate graph of the form given in Figure 1.7. Imagine that $X_{t}$ represents the total number of individuals in a population at time $t$. Jumps to the right correspond to births, and jumps to the left to deaths. The birth rates $\left{b_{i}\right}$ and the death rates $\left{d_{i}\right}$ may differ from state to state. Many applications of Markov chains involve processes of this kind. Note that the process jumps from one state to

the next according to a Markov chain with transition probabilities $K_{0,1}=1$, $K_{i, i+1}=b_{i} /\left(b_{i}+d_{i}\right)$, and $K_{i, i-1}=d_{i} /\left(b_{i}+d_{i}\right), i=1,2, \ldots$. Moreover, it spends an $\operatorname{Exp}\left(b_{0}\right)$ amount of time in state 0 and $\operatorname{Exp}\left(b_{i}+d_{i}\right)$ in the other states.
Limiting Behavior We now formulate the continuous-time analogues of (1.34) and Theorem 1.13.2. Irreducibility and recurrence for Markov jump processes are defined in the same way as for Markov chains. For simplicity, we assume that $\mathscr{E}={1,2, \ldots}$. If $X$ is a recurrent and irreducible Markov jump process, then regardless of $i$,
$$\lim {t \rightarrow \infty} \mathbb{P}\left(X{t}=j \mid X_{0}=i\right)=\pi_{j}$$
for some number $\pi_{j} \geqslant 0$. Moreover, $\pi=\left(\pi_{1}, \pi_{2}, \ldots\right)$ is the solution to
$$\sum_{j \neq i} \pi_{i} q_{i j}=\sum_{j \neq i} \pi_{j} q_{j i}, \quad \text { for all } i=1, \ldots, m$$
with $\sum_{j} \pi_{j}=1$, if such a solution exists, in which case all states are positive recurrent. If such a solution does not exist, all $\pi_{j}$ are 0 .

As in the Markov chain case, $\left{\pi_{j}\right}$ is called the limiting distribution of $X$ and is usually identified with the row vector $\pi$. Any solution $\pi$ of (1.42) with $\sum_{j} \pi_{j}=1$ is called a stationary distribution, since taking it as the initial distribution of the Markov jump process renders the process stationary.

## 统计代写|蒙特卡洛方法代写monte carlo method代考|GAUSSIAN PROCESSES

The normal distribution is also called the Gaussian distribution. Gaussian processes are generalizations of multivariate normal random vectors (discussed in Section 1.10). Specifically, a stochastic process $\left{X_{t}, t \in \mathscr{T}\right}$ is said to be Gaussian if all its finite-dimensional distributions are Gaussian. That is, if for any choice of $n$ and $t_{1}, \ldots, t_{n} \in \mathscr{T}$, it holds that
$$\left(X_{t_{1}}, \ldots, X_{t_{n}}\right)^{\top} \sim \mathrm{N}(\boldsymbol{\mu}, \Sigma)$$
for some expectation vector $\boldsymbol{\mu}$ and covariance matrix $\Sigma$ (both of which depend on the choice of $\left.t_{1}, \ldots, t_{n}\right)$. Equivalently, $\left{X_{t}, t \in \mathscr{T}\right}$ is Gaussian if any linear combination $\sum_{i=1}^{n} b_{i} X_{t_{i}}$ has a normal distribution. Note that a Gaussian process is determined completely by its expectation function $\mu_{t}=\mathbb{E}\left[X_{t}\right], t \in \mathscr{T}$, and covariance function $\Sigma_{s, t}=\operatorname{Cov}\left(X_{s}, X_{t}\right), s, t \in \mathscr{T}$.

The Wiener process can be defined as a Gaussian process $\left{X_{t}, t \geqslant 0\right}$ with expectation function $\mu_{t}=0$ for all $t$ and covariance function $\Sigma_{s, t}=s$ for $0 \leqslant s \leqslant t$. The Wiener process has many fascinating properties (e.g., [11]). For example, it is a Markov process (i.e., it satisfies the Markov property $(1.30)$ ) with continuous sample paths that are nowhere differentiable. Moreover, the increments $X_{t}-X_{s}$ over intervals $[s, t]$ are independent and normally distributed. Specifically, for any $t_{1}<t_{2} \leqslant t_{3}<t_{4}$,
$$X_{t_{4}}-X_{t_{3}} \quad \text { and } \quad X_{t_{2}}-X_{t_{1}}$$
are independent random variables, and for all $t \geqslant s \geqslant 0$,
$$X_{t}-X_{s} \sim \mathrm{N}(0, t-s) .$$
This leads to a simple simulation procedure for Wiener processes, which is discussed in Section 2.8.

## 统计代写|蒙特卡洛方法代写monte carlo method代考|Markov Jump Processes

q一世j=林吨↓0磷吨(一世,j)吨.

1. 鉴于它的过去，概率X从当前状态跳转一世陈述j是ķ一世j=q一世j/q一世.
2. 的时间量X在州花费j具有均值的指数分布1/qj，独立于其过去的历史。

## 统计代写|蒙特卡洛方法代写monte carlo method代考|Birth-and-Death Process

∑j≠一世圆周率一世q一世j=∑j≠一世圆周率jqj一世, 对全部 一世=1,…,米

## 统计代写|蒙特卡洛方法代写monte carlo method代考|GAUSSIAN PROCESSES

(X吨1,…,X吨n)⊤∼ñ(μ,Σ)

X吨4−X吨3 和 X吨2−X吨1

X吨−Xs∼ñ(0,吨−s).

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