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

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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|Forward and Backward Recurrence Time

Let $F_{0}$ be the cumulative distribution function (c.d.f.) of $T_{1}$ under $P_{N}^{0}$, that is
$$F_{0}(x)=P_{N}^{0}\left(T_{1} \leq x\right)$$
Taking $A=\left{T_{1}>v,-T_{0}>w\right}$ in (1.2.26), with $v, w \in \mathbb{R}{+}$, and using the $P{N}^{0}$-a.s. relations $-T_{0} \circ \theta_{t}=t$ and $T_{1} \circ \theta_{t}=T_{1}-t$, which hold true for all $t \in\left[0, T_{1}\right)$, we obtain
$$P\left(T_{1}>v,-T_{0}>w\right)=\lambda \int_{v+w}^{\infty}\left(1-F_{0}(u)\right) d u .$$
In particular, taking $v=0$ and $w=0$, we obtain that $P\left(T_{1}>0,-T_{0}>0\right)=1$ since $\lambda \int_{0}^{\infty}\left(1-F_{0}(u)\right) d u=\lambda E_{N}^{0}\left[T_{1}\right]=1$.
Taking now $v=0$, we obtain

$$P\left(-T_{0}>w\right)=\lambda \int_{w}^{\infty}\left(1-F_{0}(u)\right) d u$$
Similarly
$$P\left(T_{1}>v\right)=\lambda \int_{v}^{\infty}\left(1-F_{0}(u)\right) d u .$$
Thus $-T_{0}$ and $T_{1}$ are identically distributed under $P$. The c.d.f.
$$F(x)=\lambda \int_{0}^{x}\left(1-F_{0}(u)\right) d u, \quad x \geq 0$$
is often referred to as the excess distribution of the c.d.f. $F_{0}$.

## 金融代写|鞅论及其在金融中的应用代写Martingale theory代考|The Slivnyak Inverse Construction

Let $\left{\theta_{t}\right}, t \in \mathbb{R}$, be a flow on $(\Omega, \mathcal{F})$, and $N$ be a point process which is compatible with $\left{\theta_{t}\right}$. Let $P^{0}$ be a probability measure on $(\Omega, \mathcal{F})$ such that
$$P^{0}\left(\Omega_{0}\right)=1,$$
where $\Omega_{0}=\left{T_{0}=0\right}$. Suppose that $P^{0}$ is $\theta_{T_{n}}$-invariant:
$$P^{0}\left(\theta_{T_{n}} \in .\right)=P^{0}(.), \quad n \in \mathbb{Z} .$$
Moreover assume that the following three properties hold:
(i) $00\right]=1$,
(iii) $E^{0}[N(0, t]]<\infty, \quad \forall t0$.
We shall see that $P^{0}$ is then the Palm probability $P_{N}^{0}$ associated with the stationary point process $\left(N, \theta_{t}, P\right)$, for some probability $P$ which is $\theta_{t}$-invariant for all $t \in \mathbb{R}$. Moreover, in view of the inversion formula, $P$ will be unique.
As required by the inversion formula, if such a $P$ exists, it should satisfy
$$P(A)=\frac{1}{E^{0}\left[T_{1}\right]} E^{0}\left[\int_{0}^{T_{1}}\left(1_{A} \circ \theta_{t}\right) d t\right], \quad A \in \mathcal{F}$$
Clearly (1.3.19) defines a probability $P$ on $(\Omega, \mathcal{F})$. We must show that $P$ is $\theta_{t}$-invariant for all $t \in \mathbb{R}$, and that
(1.3.20)
$$P_{N}^{0}=P^{0}$$

## 金融代写|鞅论及其在金融中的应用代写Martingale theory代考|Other Inversion Formulas

The inversion formula (1.2.25) receives an interesting interpretation when written in the form
$$E[f]=E_{N}^{0}\left[\lambda T_{1} \frac{1}{T_{1}} \int_{0}^{T_{1}} f \circ \theta_{t} d t\right]=E_{N}^{0}\left[\lambda T_{1} f \circ \theta_{V}\right]$$
where $V$ is a random variable which, ‘conditionally upon everything else’, is uniformly distributed on $\left[0, T_{1}\right]$ (for the above to make sense, we must of course enlarge the probability space). This interpretation provides an explicit construction of $P$ from $P_{N}^{0}$. First construct the probability $P_{0}^{\prime}$ by
$(1.3 .23)$
$$d P_{0}^{\prime}=\left(\lambda T_{1}\right) d P_{N}^{0} .$$

Since $P_{N}^{0}\left(T_{0}=0\right)=1$ and $P_{0}^{\prime}$ is absolutely continuous with respect to $P_{N}^{0}$, $P_{0}^{\prime}\left(T_{0}=0\right)=1$. The stationary probability $P$ is then obtained by placing the origin at random in the interval $\left[0, T_{1}\right]$, that is
$$E[f]=E_{0}^{\prime}\left[f \circ \theta_{V}\right]$$
This construction seems to suggest that only the distributions of $T_{0}$ and $T_{1}$ are changed when passing from the Palm to the stationary probability. This is of course not true, since these two points may condition the distribution of other random variables. What is true is that, conditionally on $T_{0}$ and $T_{1}$, $P$ and $P_{N}^{0}$ are the same. In particular, if $\mathcal{G}$ is a sub $\sigma$-field of $\mathcal{F}$ such that $\mathcal{G}$ and $N$ are $P$ (resp. $P_{N}^{0}$ )-independent, then $\mathcal{G}$ and $N$ are also $P_{N}^{0}$ (resp. $P)$-independent.

In relation to what precedes, we mention yet another relation between $P$ and its Palm probability: for all $A \in \mathcal{F}$,
$$P\left(\theta_{T_{0}}^{-1} A\right)=\lambda E_{N}^{0}\left[T_{1} 1_{A}\right] .$$
The proof is immediate since this is just equality (1.3.22).

## 金融代写|鞅论及其在金融中的应用代写Martingale theory代考|Forward and Backward Recurrence Time

F0(X)=磷ñ0(吨1≤X)

F(X)=λ∫0X(1−F0(在))d在,X≥0

## 金融代写|鞅论及其在金融中的应用代写Martingale theory代考|The Slivnyak Inverse Construction

（i）00\右]=100\右]=1,
(iii)和0[ñ(0,吨]]<∞,∀吨0.

(1.3.20)

(1.3.23)
d磷0′=(λ吨1)d磷ñ0.

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