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# 数学代写|矩阵方法代写Applied Matrix Theory代考|MATH4500 Discrete Phase-Type Distributions

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## 数学代写|矩阵方法代写Applied Matrix Theory代考|Discrete Phase-Type Distributions

Let $\left{X_n\right}_{n \in \mathbb{N}}$ be a Markov chain with state space ${1,2, \ldots, p, p+1}$, where the states $1,2, \ldots, p$ are transient, and consequently, state $p+1$ is absorbing. Then $\left{X_n\right}_{n \in \mathbb{N}}$ has a transition matrix $\boldsymbol{P}$ of the form
$$\boldsymbol{P}=\left(\begin{array}{ll} \boldsymbol{T} & \boldsymbol{t} \ \mathbf{0} & 1 \end{array}\right)$$

where $T$ is a $p \times p$ subtransition matrix (i.e., a matrix of nonnegative numbers in which the rows sum to numbers less than or equal to one, written as $\boldsymbol{T e} \leq \boldsymbol{e}$ ), and $\boldsymbol{t}$ is a p-dimensional column vector. Since $t_i$ is the probability of jumping to an absorbing state directly from state $i$, we shall refer to these probabilities as exit probabilities. Since the rows sum to 1 , we must have that
$$\boldsymbol{t}=\boldsymbol{e}-\boldsymbol{T e}=(\boldsymbol{I}-\boldsymbol{T}) \boldsymbol{e},$$
where $\boldsymbol{e}^{\prime}=(1,1, \ldots, 1)$ is the column vector of ones. Thus $\boldsymbol{t}$ can be obtained from $\boldsymbol{T}$ and hence discarded when the necessary parameters are specified. Let $\pi_i=$ $\mathbb{P}\left(X_0=i\right), \boldsymbol{\pi}=\left(\pi_1, \ldots, \pi_p\right)$ and assume that $\boldsymbol{\pi} \boldsymbol{e}=\pi_1+\cdots+\pi_p=1$.

Definition 1.2.54. Let $\tau=\inf \left{n \geq 1 \mid X_n=p+1\right}$ be the time until absorption. Then we say that $\tau$ has a (discrete) phase-type distribution with initial distribution $\pi$ and subtransition matrix $\boldsymbol{T}$, and we write
$$\tau \sim \mathrm{DPH}_p(\boldsymbol{\pi}, \boldsymbol{T})$$

## 数学代写|矩阵方法代写Applied Matrix Theory代考|Markov Jump Processes

In this section we consider Markov processes in continuous time that take values in a discrete (finite or at most countable) state space. By nature, such processes are piecewise constant, and transitions occur via jumps. They are often referred to as Markov jump processes or continuous-time Markov chains. Which value the process takes at the time of a jump can be assigned arbitrarily, however, we will always assume that the process takes the value of the state to which it jumps. This assumption makes Markov jump processes continuous from the right (and with limits from the left), i.e., they are so-called càdlàg processes.

Definition 1.3.1. A continuous-time stochastic process $\left{X_t\right}_t \geq 0$ taking values in a countable set $E$ is called a Markov jump process with state space $E$ if for all $t_n>$ $t_{n-1}>\cdots>t_1>0$ and $i_n, i_{n-1}, \ldots, i_0 \in E$, we have that
$$\mathbb{P}\left(X_{t_n}=i_n \mid X_{t_{n-1}}=i_{n-1}, \ldots, X_{t_1}=i_1, X_0=i_0\right)=\mathbb{P}\left(X_{t_n}=i_n \mid X_{t_{n-1}}=i_{n-1}\right) .$$
The process is called time-homogeneous if the transition probabilities $\mathbb{P}\left(X_{t+h}=\right.$ $\left.j \mid X_t=i\right)$ depend only on $h$, in which case it is denoted by $p_{i j}(h)$ and referred to as an $h$-step transition probability. Throughout, we assume that all Markov jump processes are time-homogeneous.
The transition probabilities are then arranged in transition matrices
$$\boldsymbol{P}(h)=\left{p_{i j}(h)\right}_{i, j \in E}, \quad h \geq 0 .$$

## 数学代写|矩阵方法代写Applied Matrix Theory代考|Discrete Phase-Type Distributions

$$\boldsymbol{P}=\left(\begin{array}{lll} \boldsymbol{T} & \boldsymbol{t} 0 & 1 \end{array}\right)$$

$$t=e-T e=(I-T) e,$$
$\mathbb{P}\left(X_0=i\right), \pi=\left(\pi_1, \ldots, \pi_p\right)$ 并假设 $\pi e=\pi_1+\cdots+\pi_p=1$.

$$\tau \sim \operatorname{DPH}p(\pi, \boldsymbol{T})$$

## 数学代写|矩阵方法代写Applied Matrix Theory代考|Markov Jump Processes

$$\mathbb{P}\left(X_{t_n}=i_n \mid X_{t_{n-1}}=i_{n-1}, \ldots, X_{t_1}=i_1, X_0=i_0\right)=\mathbb{P}\left(X_{t_n}=i_n \mid X_{t_{n-1}}=i_{n-1}\right) .$$

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