金融代写|鞅论及其在金融中的应用代写Martingale theory代考|Palm Theory in Discrete Time

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

金融代写|鞅论及其在金融中的应用代写Martingale theory代考|Palm Theory in Discrete Time

The Palm theory in continuous time has a counterpart in discrete time which can be developed in very elementary terms. In the present section, we shall briefly sketch this discrete time version, leaving the details to the reader. For this, we shall first adapt the notation and definitions used in continuous time to the discrete time situation.

In discrete time, a simple point process on $\mathbb{Z}$ is just a sequence $\left{U_{n}\right}, n \in \mathbb{Z}$, where $U_{n}=0$ or 1 . It is called stationary if the sequence $\left{U_{n}\right}$ is strictly stationary, and its intensity is then defined by
$$\lambda_{U}=E\left[U_{0}\right]$$
Observe that $0 \leq \lambda_{U} \leq 1$.
The canonical framework in discrete time is the following: $(\Omega, \mathcal{F})$ is a measurable space endowed with an invertible measurable map $\theta_{1}:(\Omega, \mathcal{F}) \rightarrow$ $(\Omega, \mathcal{F})$ such that $\theta_{1}^{-1}$ is measurable (we can think of $\theta_{1}$ as the shift to the left, although this is not necessary). Define $\theta_{n}=\theta_{1}^{n}$ for all $n \in \mathbb{Z}$. A probability $P$ on $(\Omega, \mathcal{F})$ such that
$$P \circ \theta_{1}^{-1}=P$$
( $P$ is $\theta_{1}$-invariant) is called a stationary probability.
A sequence $\left{Z_{n}\right}, n \in \mathbb{Z}$, of random elements with values in an arbitrary measurable space $(E, \mathcal{E})$ is said to be compatible with $\left{\theta_{n}\right}$ if
$$Z_{n}(\omega)=Z_{0}\left(\theta_{n} \omega\right)$$
for all $n \in \mathbb{Z}$ (with the convention that $\theta_{1}^{0}$ is the identity). The sequence $\left{Z_{n}\right}$ is then strictly stationary (with respect to $P$ ). Thus, if the point process $\left{U_{n}\right}$ is compatible with $\left{\theta_{n}\right}$, it is stationary. We shall assume it is so. The Palm probability $P_{U}$ associated with $\left{U_{n}\right}$ is defined by the formula.

金融代写|鞅论及其在金融中的应用代写Martingale theory代考|Stochastic Intensity

Roughly speaking, Palm probability tells us what happens when there is a point at time $t$. The concept of stochastic intensity introduced in the present chapter represents in some way a complementary point of view: it is concerned with the expectation of seeing a point at time $t$ (in a small interval after $t$ ) knowing the past history of the point process.

The connection between the two points of view will be formalized in $\S$ 1.9. Before giving the definition of stochastic intensity, we must spend some time introducing notation and a few definitions from the theory of stochastic processes. For all the statements announced without proof in the following sections, the reader is referred to the monograph: Brémaud (1981) Point Processes and Queues, Springer-Verlag, New York.

金融代写|鞅论及其在金融中的应用代写Martingale theory代考|Stochastic Intensity Kernel

Let $N$ be a simple point process, not necessarily stationary, let $\left{\mathcal{F}{t}\right}$ be a history of $N$, and let ${\lambda(t)}$ be a non-negative measurable process adapted to $\left{\mathcal{F}{t}\right}$. The process ${\lambda(t)}$ is called an $\mathcal{F}{t}$-intensity of $N$ if it is locally integrable (i.e. $\int{C} \lambda(s) d s<\infty$, for all bounded Borel sets $C$ ) and if
$$E\left[N((a, b]) \mid \mathcal{F}{a}\right]=E\left[\int{a}^{b} \lambda(s) d s \mid \mathcal{F}{a}\right],$$ for all $(a, b] \in \mathcal{B}$. Without loss of generality, the stochastic intensity ${\lambda(t)}$ can be assumed to be $\mathcal{F}{t}$-predictable (cf. Brémaud (1981), Chapter II, T12, p. 31).

Example 1.8.2. The Poisson process. Let $N$ be a Poisson process with associated intensity measure
$$E[N(C)]=\int_{C} \lambda(s) d s,$$
where ${\lambda(t)}$ is a deterministic locally integrable function. Then clearly, in view of the independence of the increments of a Poisson process, ${\lambda(t)}$ is the $\mathcal{F}_{t}^{N}$-intensity of $N$.

Example 1.8.3. Markov chains. Let ${X(t)}, t \in \mathbb{R}{+}$, be a regular jump Markov chain ([39]) taking its values in a countable state space $\mathcal{E}$ and corlol sample paths. Then the following limits exist and belong to $\mathbb{R}{+}$:
$$q_{i}=\lim {h \rightarrow 0} \frac{1-p{i i}(h)}{h} \text { and } q_{i j}=\lim {h \rightarrow 0} \frac{p{i j}(h)}{h}, i \neq j .$$
Moreover
$$q_{i}<\infty \text { and } \sum_{\substack{j \in \mathcal{E} \ j \neq i}} q_{i j}=q_{i} .$$
Let $N_{i j}$ be the point process counting the transitions from $i$ to $j$, i.e. for $C \in \mathcal{B}, N_{i j}(C)=\sum_{s \in C} 1_{X(s-)=i, X(s)=j}$. It can be shown (this is essentially Lévy’s formula for Markov chains (see [36]) that $N_{i j}$ admits the $\mathcal{F}{t}^{X}$-intensity $\left{\lambda{i j}(t)\right}=\left{q_{i j} 1_{X(t)=i}\right}$.

Let $N$ be a point process compatible with the flow $\left{\theta_{t}\right}$ and let $\left{Z_{n}\right}$ be a sequence of marks of $\left(N, \theta_{t}\right),(K, \mathcal{K})$ being the corresponding mark space. Let $\left{\mathcal{F}{t}\right}$ be a history of the marked point process $\left(N,\left{Z{n}\right}\right)$. Let $\lambda(t, \omega, L), t \in$ $\mathbb{R}, \omega \in \Omega, L \in \mathcal{K}$, be a stochastic intensity kernel, that is:

• For all $t, \omega, \lambda(t, \omega, .)$ is a measure on $(K, \mathcal{K})$.
• For all $L \in \mathcal{K},{\lambda(t, L)}$ is adapted to $\left{\mathcal{F}_{t}\right}$, where $\lambda(t, L)(\omega) \stackrel{\text { def }}{=} \lambda(t, \omega, L)$.

金融代写|鞅论及其在金融中的应用代写Martingale theory代考|Palm Theory in Discrete Time

λ在=和[在0]

( 磷是θ1-invariant) 称为平稳概率。

金融代写|鞅论及其在金融中的应用代写Martingale theory代考|Stochastic Intensity Kernel

q一世=林H→01−p一世一世(H)H 和 q一世j=林H→0p一世j(H)H,一世≠j.

q一世<∞ 和 ∑j∈和 j≠一世q一世j=q一世.

• 对全部吨,ω,λ(吨,ω,.)是衡量(ķ,ķ).
• 对全部大号∈ķ,λ(吨,大号)适应\left{\mathcal{F}_{t}\right}\left{\mathcal{F}_{t}\right}， 在哪里λ(吨,大号)(ω)= 定义 λ(吨,ω,大号).

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