### 金融代写|鞅论及其在金融中的应用代写Martingale theory代考|Palm Probability

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• Statistical Inference 统计推断
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• 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|The Matthes Definition in Terms of Counting

Let $\widetilde{Z}$ be the sequence of universal marks $(\S 1.1 .3)$ associated with $\left(N, \theta_{t}, P\right)$ and denote by $\tilde{\lambda}$ the Campbell measure $\lambda_{\tilde{Z}}$ associated with $\left((N, \widetilde{Z}), \theta_{t}, P\right)$. Also denote $\nu_{\bar{Z}}$ (see $\left(1.1 .25\right.$ ) for the definition of $\nu_{Z}$ ) by $P_{N}^{0}$. Thus, recalling that the marks $\bar{Z}{n}=\theta{T_{n}}$ take their values in $(\Omega, \mathcal{F})$, we see that $P_{N}^{0}$ is a probability measure on $(\Omega, \mathcal{F})$ defined by
\begin{aligned} P_{N}^{0}(A) &=\frac{1}{\lambda l(C)} E\left[\sum_{n \in Z} 1_{A}\left(\theta_{T_{n}}\right) 1_{C}\left(T_{n}\right)\right] \ &=\frac{1}{\lambda l(C)} E\left[\int_{C}\left(1_{A} \circ \theta_{s}\right) N(d s)\right], \quad A \in \mathcal{F} \end{aligned}
It is called the Palm probability of the stationary point process $\left(N, \theta_{t}, P\right)$. We immediately observe that
$$P_{N}^{0}\left(T_{0}=0\right)=1 .$$
This equality follows when taking $A=\left{T_{0}=0\right}$ in (1.2.1) and observing that
\begin{aligned} \int_{C}\left(1_{\left{T_{0}=0\right}} \circ \theta_{s}\right) N(d s) &\left.=\sum_{n \in Z} 1_{\left{T_{0} \circ \theta_{T_{n}}\right.}=0\right} \ &=\sum_{n \in Z} 1_{C}\left(T_{n}\right) \end{aligned}
since $T_{0} \circ \theta_{T_{n}} \equiv 0$.
Let now $Z$ be an arbitrary sequence of marks of $\left(N, \theta_{t}\right)$ with values in $(K, \mathcal{K})$, and let $L \in \mathcal{K}$. Taking $A=\left{Z_{0} \in L\right}$ in (1.2.1) and observing that $1_{\left{Z_{0} \in L\right}} \circ \theta_{T_{n}}=1_{\left{Z_{0} \circ \theta_{T_{n}} \in L\right}}=1_{\left{Z_{n} \in L\right}}$, we see that
$$\nu_{Z}(L)=P_{N}^{0}\left(Z_{0} \in L\right)$$
Thus formula (1.1.26) takes the form
$$\lambda_{Z}(C \times L)=\lambda l(C) P_{N}^{0}\left(Z_{0} \in L\right)$$
In order to gain some intuition as to what is going on, we now give complementary points of view on (1.2.1).

## 金融代写|鞅论及其在金融中的应用代写Martingale theory代考|lnvariance of Palm Probability

The Palm probability $P_{N}^{0}$ of the stationary point process $\left(N, \theta_{t}, P\right)$ has its mass concentrated on $\Omega_{0}=\left{T_{0}=0\right}$. Defining
$$\theta=\theta_{T_{1}}$$
we see that $\theta$ is a bijection of $\Omega_{0}$, with inverse $\theta^{-1}=\theta_{T_{-1}}$, and that
$$\theta_{T_{n}}=\theta^{n} \text { on } \Omega_{0}, \text { for all } n \in \mathbb{Z} \text {. }$$
The following result is expected in view of the heuristic interpretation given in the previous subsection:

$(1.2 .16)$
$P_{N}^{0}$ is $\theta$-invariant.
Proof: Take $A \in \mathcal{F}, A \subset \Omega_{0}$
$$\left|P_{N}^{0}(A)-P_{N}^{0}\left(\theta^{-1}(A)\right)\right| \leq \frac{1}{\lambda t} E\left|\sum_{n \in \mathbb{Z}}\left(1_{A} \circ \theta_{T_{n}}-1_{\theta^{-1}(A)} \circ \theta_{T_{n}}\right) 1_{(0, t]}\left(T_{n}\right)\right|$$
where we have applied the defining formula (1.2.1) with $C=(0, t]$. Since $1_{\theta^{-1}(A)} \circ \theta_{T_{n}}=1_{A} \circ \theta_{T_{n+1}}$, we see that
$$\left|P_{N}^{0}(A)-P_{N}^{0}\left(\theta^{-1}(A)\right)\right| \leq \frac{1}{\lambda t} .$$
Now letting $t \rightarrow \infty$, we obtain
$$P_{N}^{0}(A)=P_{N}^{0}\left(\theta^{-1}(A)\right)$$
for all $A \in \mathcal{F}, A \subset \Omega_{0}$.
So if ${Z(t)}$ is compatible with the flow $\left{\theta_{t}\right}$ (or equivalently if it is a (time)-stationary process under $P$ ), then the sequence $\left{Z\left(T_{n}\right)\right}$ is a stationary sequence under $P_{N}^{0}$.

## 金融代写|鞅论及其在金融中的应用代写Martingale theory代考|Mecke’s Formula

Let $v$ be the real-valued function defined on $\Omega \times \mathbb{R}$ by
$$v(\omega, t)=1_{A}(\omega) 1_{C}(t)$$
where $A \in \mathcal{F}, C \in \mathcal{B}$. By definition of the product measures $P_{N}^{0}(d \omega) \times d t$ and $P(d \omega) N(\omega, d t)$, formula (1.2.1) reads
$$\lambda \iint_{\Omega \times \mathbb{R}} v(\omega, t) P_{N}^{0}(d \omega) d t=\iint_{\Omega \times \mathbb{R}} v\left(\theta_{t} \omega, t\right) P(d \omega) N(\omega, d t) .$$
By standard monotone class arguments, $(1.2 .17)$ remains true for all nonnegative measurable functions $v$ from $(\Omega \times \mathbb{R}, \mathcal{F} \otimes \mathcal{B})$ into $(\mathbb{R}, \mathcal{B})$.

Formula (1.2.17) is known as the generalized Campbell formula. In this book, it will be called Mecke’s formula, after its author.

The original Campbell formula is obtained by specializing Mecke’s formula $(1.2 .17)$ to
$$v(\omega, t)=f\left(t, Z_{0}(\omega)\right),$$
where $\left{Z_{n}\right}$ is a sequence of marks of $\left(N, \theta_{t}\right)$, with values in $(K, \mathcal{K})$. In view of $(1.2 .4)$ :

$$\lambda \iint_{\Omega \times \mathbb{R}} f\left(t, Z_{0}(\omega)\right) P_{N}^{0}(d \omega) d t=\iint_{\mathbb{R} \times K} f(t, z) \lambda_{Z}(d t \times d z)$$
On the other hand
\begin{aligned} \iint_{\Omega \times \mathbb{R}} f\left(t, Z_{0}\left(\theta_{t} \omega\right)\right) P(d \omega) N(\omega, d t) &=E\left[\sum_{n \in \mathcal{Z}} f\left(T_{n}, Z_{0}\left(\theta_{T_{n}}\right)\right)\right] \ &=E\left[\sum_{n \in \mathcal{Z}} f\left(T_{n}, Z_{n}\right)\right] . \end{aligned}
Therefore (Campbell’s formula)
$$E\left[\sum_{n \in Z} f\left(T_{n}, Z_{n}\right)\right]=\iint_{\mathbb{R} \times K} f(t, z) \lambda_{Z}(d t \times d z)$$
for all non-negative measurable functions $f$ from $(\mathbb{R} \times K, \mathcal{B} \times \mathcal{K})$ into $(\mathbb{R}, \mathcal{B})$.

## 金融代写|鞅论及其在金融中的应用代写Martingale theory代考|The Matthes Definition in Terms of Counting

\begin{对齐} \int_{C}\left(1_{\left{T_{0}=0\right}} \circ \theta_{s}\right) N(d s) &\left.=\sum_{ n \in Z} 1_{\left{T_{0} \circ \theta_{T_{n}}\right.}=0\right} \ &=\sum_{n \in Z} 1_{C}\left (T_{n}\right) \end{对齐}\begin{对齐} \int_{C}\left(1_{\left{T_{0}=0\right}} \circ \theta_{s}\right) N(d s) &\left.=\sum_{ n \in Z} 1_{\left{T_{0} \circ \theta_{T_{n}}\right.}=0\right} \ &=\sum_{n \in Z} 1_{C}\left (T_{n}\right) \end{对齐}

ν从(大号)=磷ñ0(从0∈大号)

λ从(C×大号)=λl(C)磷ñ0(从0∈大号)

## 金融代写|鞅论及其在金融中的应用代写Martingale theory代考|lnvariance of Palm Probability

θ=θ吨1

θ吨n=θn 在 Ω0, 对全部 n∈从.

(1.2.16)

|磷ñ0(一种)−磷ñ0(θ−1(一种))|≤1λ吨和|∑n∈从(1一种∘θ吨n−1θ−1(一种)∘θ吨n)1(0,吨](吨n)|

|磷ñ0(一种)−磷ñ0(θ−1(一种))|≤1λ吨.

## 金融代写|鞅论及其在金融中的应用代写Martingale theory代考|Mecke’s Formula

λ∬Ω×R在(ω,吨)磷ñ0(dω)d吨=∬Ω×R在(θ吨ω,吨)磷(dω)ñ(ω,d吨).

∬Ω×RF(吨,从0(θ吨ω))磷(dω)ñ(ω,d吨)=和[∑n∈从F(吨n,从0(θ吨n))] =和[∑n∈从F(吨n,从n)].

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