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

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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|Ergodicity of a Flow

Discrete Flows. Let $\left(\Omega, \mathcal{F}, P^{0}\right)$ be a probability space and let $\theta$ be discrete flow, that is, a bijective and measurable map from $\Omega$ to itself, which preserves $P^{0}$, that is, $P^{0} \circ \theta=P^{0}$.

An event $A \in \mathcal{F}$ is said to be strictly invariant if $\theta^{-1} A=A$ and invariant if $P^{0}\left(A \Delta \theta^{-1} A\right)=0$, where $\Delta$ denotes the symmetrical difference. It is said to be $\theta$-contracting if $P^{0}\left(A^{c} \cap \theta^{-1} A\right)=0$.

Since the events $A$ and $\theta^{-1} A$ have the same probability, all $\theta$-contracting events are $\theta$-invariant.

Also notice that for all $\theta$-invariant events $A$, the event $B=\limsup _{n} \theta^{-n} A$ is strictly $\theta$-invariant and such that $P^{0}(A)=P^{0}(B)$. So, for all invariant events, there exists a strictly invariant event with the same probability.
The discrete flow $\theta$ is ergodic if all $\theta$-invariant events are of probability either 0 or 1 .

In view of the last observation, $\theta$ is ergodic if and only if all strictly $\theta$-invariant events are of probability either 0 or 1 .

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

Let $\left(N, \theta_{t}, P\right)$ be a stationary point process with non-zero and finite intensity and let $P_{N}^{0}$ be the associated Palm probability. Denote $\theta_{T_{1}}$ by $\theta$.

Property 1.6.1. Let $A \in \mathcal{F}$ be a $\left{\theta_{t}\right}$-invariant event. Then $P(A)=1$ if and only if $P_{N}^{0}(A)=1$.
Proof: If $P_{N}^{0}(A)=1$, the inversion formula $(1.2 .26)$ gives

\begin{aligned} P(A) &=\lambda \int_{0}^{\infty} P_{N}^{0}\left(u<T_{1}, \theta_{-u} A\right) d u \ &=\lambda \int_{0}^{\infty} P_{N}^{0}\left(u<T_{1}, A\right) d u \quad\left(\theta_{t} \text {-invariance of } A\right) \ &=\lambda \int_{0}^{\infty} P_{N}^{0}\left(u<T_{1}\right) d u \quad\left(P_{N}^{0}(A)=1\right) \ &=\lambda E_{0}^{0}\left[T_{1}\right]=1 \end{aligned}
Conversely, supposing that $P(A)=1$,
$$1=P(A)=\lambda \int_{0}^{\infty} P_{N}^{0}\left(u<T_{1}, A\right) d u$$
and therefore, since $1=\lambda \int_{0}^{\infty} P_{N}^{0}\left(u<T_{1}\right) d u$,
$$0=\lambda \int_{0}^{\infty} P_{N}^{0}\left(u<T_{1}, \bar{A}\right) d u .$$
This implies $P_{N}^{0}\left(u<T_{1}, \bar{A}\right)=0, d u$-almost everywhere, from which we conclude that $P_{N}^{0}(A)=1$ (recall that $T_{1}<\infty$ a.s. since its mean is finite).

## 金融代写|鞅论及其在金融中的应用代写Martingale theory代考|Ergodicity Under the Stationary Probability

Let $\left(N, \theta_{t}, P\right)$ be a stationary point process with a finite intensity $\lambda$, and let $P_{N}^{0}$ be its associated Palm probability. Denote $\theta_{T_{1}}=\theta$.
Property 1.6.3. $\left(P, \theta_{t}\right)$ is ergodic if and only if $\left(P_{N}^{0}, \theta\right)$ is ergodic.
Proof: Suppose for instance that $\left(P, \theta_{t}\right)$ is ergodic and that $\left(P_{N}^{0}, \theta\right)$ is not. Then there must be a decomposition of the type (1.6.4). Let $P_{1}$ and $P_{2}$ be the stationary probabilities associated with $Q_{1}$ and $Q_{2}$ (see $\S 1.3 .5$ ). The inversion formula applied to (1.6.3) gives
$$P=\alpha_{1} \frac{\lambda}{\lambda_{1}} P_{1}+\alpha_{2} \frac{\lambda}{\lambda_{2}} P_{2}$$
where $\frac{1}{\lambda}=E_{N}^{0}\left[T_{1}\right], \frac{1}{\lambda_{1}}=E_{Q_{1}}\left[T_{1}\right], \frac{1}{\lambda_{2}}=E_{Q_{2}}\left[T_{1}\right]$ (note that $\lambda_{1}$ and $\lambda_{2}$ must be strictly positive). Therefore $\left(\theta_{t}, P\right)$ is not ergodic, hence a contradiction. The proof of the converse part is based on the observation that (1.6.6) implies
$$P_{N}^{0}=\beta_{1} \frac{\lambda_{1}}{\lambda} P_{1, N}^{0}+\beta_{2} \frac{\lambda_{2}}{\lambda} P_{2, N}^{0},$$
where $\lambda=E[N(0,1]], \lambda_{1}=E_{P_{1}}[N(0,1]], \lambda_{2}=E_{P_{2}}[N(0,1]]$.

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

1=磷(一种)=λ∫0∞磷ñ0(在<吨1,一种)d在

0=λ∫0∞磷ñ0(在<吨1,一种¯)d在.

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