### 金融代写|鞅论及其在金融中的应用代写Martingale theory代考|Selected Transitions of a Stationary Markov Chain

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## 金融代写|鞅论及其在金融中的应用代写Martingale theory代考|Selected Transitions of a Stationary Markov Chain

The setting and the notation of the present subsection are the same as in Example 1.1.5. The intensity $\lambda_{H}$ of $N_{H}$ is
\begin{aligned} E\left[N_{H}(0,1]\right] &=E\left[\int_{(0,1]} 1_{H}\left(X_{s-}, X_{s}\right) N(d s)\right] \ &=E\left[\int_{(0,1]} \sum_{(i, j) \in H} 1_{{i}}\left(X_{s-}\right) N_{i j}(d s)\right] \end{aligned}
where $N_{i j}=N_{{(i, j)}}$ counts the transitions from $i$ to $j$. Therefore from Lévy’s formula (see e.g. [36])
$$\lambda_{H}=\sum_{(i, j) \in H} \pi(i) q_{i j}$$
We assume that
$$0<\lambda_{H}<\infty$$
so that we can define the Palm probability $P_{N_{H}}^{0}$ associated with $N_{H}$. Let now $g: \mathcal{E} \times \mathcal{E} \rightarrow \mathbb{R}$ be non-negative and measurable. Then
\begin{aligned} E_{N_{H}}^{0}\left[g\left(X_{0-1}, X_{0}\right)\right] &=\frac{1}{\lambda_{H}} E\left[\int_{(0,1]} g\left(X_{s-}, X_{s}\right) N_{H}(d s)\right] \ &=\frac{1}{\lambda_{H}} E\left[\int_{(0,1]} \sum_{(i, j) \in H} g(i, j) 1_{{i}}\left(X_{s-}\right) N_{i j}(d s)\right] \ &=\frac{1}{\lambda_{H}} E\left[\int_{(0,1]} \sum_{(i, j) \in H} g(i, j) 1_{{i}}\left(X_{s-}\right) q_{i j} d s\right] \end{aligned}

## 金融代写|鞅论及其在金融中的应用代写Martingale theory代考|Stationary Semi-Markov Process

A semi-Markov process on the denumerable state space $\mathcal{E}$ is constructed as follows.

Let $I P=\left{p_{i j}\right}, i, j \in \mathcal{E}$, be a stochastic matrix on $\mathcal{E}$, assumed irreducible and positive recurrent (in short, ergodic). Its unique stationary distribution is denoted by $\pi={\pi(i)}, i \in \mathcal{E}$.

For each $i, j \in \mathcal{E}$, let $G_{i j}(t)$ be the cumulative distribution function of some strictly positive and proper random variable: thus $G_{i j}(0)=0$ and $G_{i j}(\infty)=$ 1. Denote by $m_{i j}$ the mean
$$m_{i j}=\int_{0}^{\infty} t G_{i j}(d t)=\int_{0}^{\infty}\left(1-G_{i j}(t)\right) d t<\infty .$$
Recall at this stage that if $U$ is a random variable uniformly distributed on $[0,1], G_{i j}^{-1}(U)$ is a random variable with c.d.f. $G_{i j}(t)$ (here $G_{i j}^{-1}$ is the inverse of $\left.G_{i j}\right)$.

Let $\left{X_{n}\right}, n \in \mathbb{Z}$, be a stationary Markov chain with transition matrix $I P$, defined on some probability space with a probability $\mathcal{P}^{0}$, and let $\left{U_{n}\right}, n \in$ $\mathbb{Z}$, be a sequence of i.i.d. random variables, defined on the same space and uniformly distributed on $[0,1]$. Assume moreover that the sequences $\left{U_{n}\right}$ and $\left{X_{n}\right}$ are independent under $\mathcal{P}^{0}$.
Define
$$S_{n}=G_{X_{n} X_{n+1}}^{-1}\left(U_{n}\right)$$
In particular, conditionally on $X_{n}=i$ and $X_{n+1}=j, S_{n}$ is distributed according to the c.d.f. $G_{i j}(t)$. Moreover, conditionally on the whole sequence $\left{X_{n}\right}$, the sequence $\left{S_{n}\right}$ forms an independent family of random variables.
We can now define a point process $N$ by
$$T_{0}=0, T_{n+1}-T_{n}=S_{n} \quad(n \in \mathbb{Z})$$

## 金融代写|鞅论及其在金融中的应用代写Martingale theory代考|Delayed Marked Point Process

Let $\left(\Omega, P, \mathcal{F}, \theta_{t}\right)$ be a stationary framework, let $N$ be a $\theta_{t}$-compatible point process with points $\left{T_{n}\right}$, and let $\left{Z_{n}\right}$ and $\left{V_{n}\right}$ be two mark sequences, with values in $(K, \mathcal{K})$ and $(\mathbb{R}, \mathcal{B})$ respectively.
Let $N_{Z}^{\prime}$ be the random measure on $\mathbb{R} \times K$ defined by
$$N_{Z}^{\prime}(C \times L)=\sum_{n \in Z} 1_{C}\left(T_{n}+V_{n}\right) 1_{L}\left(Z_{n}\right) .$$
Let $\left{T_{n}^{\prime}, Z_{n}^{\prime}\right}$, denote the sequence of points of this random measure, where the numbering obeys the usual conventions. Let $N^{\prime}$ be defined by $N^{\prime}(.)=$ $N_{Z}^{\prime}(., K)$.

So, the points of $N^{\prime}$ are obtained from those of $N$ by delaying the $n$-th point of $T_{n}$ of $V_{n}$, and the mark of the resulting point is $Z_{n}$. Note that the order of points is not assumed to be preserved by this transformation, that is $T_{n}^{\prime}$ is not necessarily $T_{n}+V_{n}$.

## 金融代写|鞅论及其在金融中的应用代写Martingale theory代考|Selected Transitions of a Stationary Markov Chain

λH=∑(一世,j)∈H圆周率(一世)q一世j

0<λH<∞

## 金融代写|鞅论及其在金融中的应用代写Martingale theory代考|Delayed Marked Point Process

ñ从′(C×大号)=∑n∈从1C(吨n+在n)1大号(从n).

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