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

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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|The Inversion Formula

The original ( $\theta_{t}$-invariant) probability can be recovered from the Palm probability. Let $h(\omega, t)$ be a real valued stochastic process such that
$$\int_{\mathbb{R}} h(\omega, t) N(\omega, d t)=1 \quad P \text {-a.s. }$$
From Mecke’s formula applied to $v(\omega, t)=h\left(\theta_{-t} \omega, t\right) f\left(\theta_{-t} \omega\right)$, it follows that for all non-negative random variables $f$,
$$E[f]=\lambda \iint_{\Omega \times \mathbb{R}} h\left(\theta_{-t} \omega, t\right) f\left(\theta_{-t} \omega\right) P_{N}^{0}(d \omega) d t .$$
When taking
$$h(\omega, t)=1_{\left[T_{0}(\omega), 0\right)}(t),$$
and when using the fact that $P_{N}^{0}$-a.s., $-T_{0} \circ \theta_{u}=u$ for all $u \in\left[0, T_{1}\right)$, we obtain the inversion formula of Ryll-Nardzewski and Slivnyak:
$$E[f]=\lambda E_{N}^{0}\left[\int_{0}^{T_{1}}\left(f \circ \theta_{t}\right) d t\right] .$$
In the special case $f=1_{A}, A \in \mathcal{F}$, this formula reads
$$P(A)=\lambda \int_{0}^{\infty} P_{N}^{0}\left(T_{1}>t, \theta_{t} \in A\right) d t .$$
Taking $f=1$ in (1.2.25) also gives:
$$\lambda E_{N}^{0}\left[T_{1}\right]=1 .$$
Exercise 1.2.2. (F) Prove that for all $n \in \mathbb{Z}$, and for all $f: \Omega \rightarrow \mathbb{R}$,
$$E[f]=\lambda E_{N}^{0}\left[\int_{T_{n}}^{T_{n+1}}\left(f \circ \theta_{t}\right) d t\right]$$
and for all $A \in \mathcal{F}$,
$$P(A)=\lambda \int_{\mathbb{R}} P_{N}^{0}\left(T_{n}<t \leq T_{n+1}, \theta_{t} \in A\right) d t$$

## 金融代写|鞅论及其在金融中的应用代写Martingale theory代考|The Mean-Value Formulas

Let $\left(N, \theta_{t}, P\right)$ be a stationary point process with finite intensity, and let $P_{N}^{0}$ be the associated Palm probability. Let $\left{Z_{t}\right}, t \in \mathbb{R}$, be a stochastic process with values in a measurable space $(K, K)$ and such that
$$Z_{t}=Z_{0} \circ \theta_{t} .$$
Then, for all non-negative measurable functions $g:(K, \mathcal{K}) \rightarrow(\mathbb{R}, \mathcal{B})$
$$E\left[g\left(Z_{0}\right)\right]=\frac{E_{N}^{0}\left[\int_{0}^{T_{1}} g\left(Z_{t}\right) d t\right]}{E_{N}^{0}\left[T_{1}\right]}$$
and
$$E_{N}^{0}\left[g\left(Z_{0}\right)\right]=\frac{E\left[\sum_{n \in \mathbb{Z}} g\left(Z_{T_{n}}\right) 1_{\left{T_{n} \in(0,1]\right}}\right]}{E\left[\sum_{n \in Z} 1_{\left{T_{n} \in(0,1]\right}}\right]}$$
These formulas just rephrase the inversion formula (1.2.25) and the definition formula $(1.2 .1)$ of $P_{N}^{0}$.

Formula (1.3.2) is well-known in the context of the theory of renewal and regenerative processes.

## 金融代写|鞅论及其在金融中的应用代写Martingale theory代考|The Neveu Exchange Formula

Let $\left(N, \theta_{t}, P\right)$ and $\left(N^{\prime}, \theta_{t}, P\right)$ be two stationary point processes with finite intensities $\lambda$ and $\lambda^{\prime}$ respectively. Note that $N$ and $N^{\prime}$ are jointly stationary, in the sense that their stationarity is relative to the same quadruple $\left(\Omega, \mathcal{F}, P, \theta_{t}\right)$.
The following formula, called the exchange formula, holds:
$$\lambda E_{N}^{0}[f]=\lambda^{\prime} E_{N^{\prime}}^{0}\left[\int_{\left(0, T_{1}^{\prime}\right]}\left(f \circ \theta_{t}\right) N(d t)\right]$$
for all non-negative measurable functions $f:(\Omega, \mathcal{F}) \rightarrow(\mathbb{R}, \mathcal{B})$. Here $T_{n}^{\prime}$ is the $n$-th point of $N^{\prime}$.

Proof: By the monotone convergence theorem, we may assume $f$ bounded (say by 1, without loss of generality). With all such $f$, we associate the function
$$g \stackrel{\text { def }}{=} \int_{\left(T_{\mathrm{D}^{\prime}}, T_{1}\right]}\left(f \circ \theta_{t}\right) N(d t)$$

For all $t \in \mathbb{R}{+}$, we have $$\int{(0, t]}\left(f \circ \theta_{s}\right) N(d s)=\int_{(0, t]}\left(g \circ \theta_{s}\right) N^{\prime}(d s)+R(t)$$
where $R(t)$ consists of two terms:
$$R(t)=\int_{\left(0, T_{+}^{\prime}(0)\right]}\left(f \circ \theta_{s}\right) N(d s)-\int_{\left(t, T_{+}^{\prime}(t)\right]}\left(f \circ \theta_{s}\right) N(d s) .$$
Here $T_{+}^{\prime}(t)$ is the first point of $N^{\prime}$ strictly larger than $t$.
For all $l>0$, define
$$f_{l}=f 1_{\left{N\left(T_{\mathrm{b}}, T_{1}{ }^{\prime}\right] \leq l-1\right}}$$
and let
$$g l=\int_{\left\langle T_{0}{ }^{\prime}, T_{1}{ }^{\prime}\right]}\left(f_{l} \circ \theta_{t}\right) N(d t) .$$
For these functions, each term in $R(t)$ is bounded by $l$, and therefore the expectations are finite. Moreover, by the $\theta_{t}$-invariance of $P$, they have the same expectations, so that $E[R(t)]=0$. We therefore have
$$E\left[\int_{(0, t]}\left(f_{l} \circ \theta_{s}\right) N(d s)\right]=E\left[\int_{(0, t]}\left(g_{l} \circ \theta_{s}\right) N^{\prime}(d s)\right]$$
This and (1.2.8) imply (1.3.4) with $f=f_{l}, g=g_{l}$. Letting $l$ go to infinity, we obtain (1.3.4).

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

∫RH(ω,吨)ñ(ω,d吨)=1磷-作为

H(ω,吨)=1[吨0(ω),0)(吨),

λ和ñ0[吨1]=1.

## 金融代写|鞅论及其在金融中的应用代写Martingale theory代考|The Mean-Value Formulas

E_{N}^{0}\left[g\left(Z_{0}\right)\right]=\frac{E\left[\sum_{n \in \mathbb{Z}} g\left(Z_ {T_{n}}\right) 1_{\left{T_{n} \in(0,1]\right}}\right]}{E\left[\sum_{n \in Z} 1_{\left {T_{n} \in(0,1]\right}}\right]}E_{N}^{0}\left[g\left(Z_{0}\right)\right]=\frac{E\left[\sum_{n \in \mathbb{Z}} g\left(Z_ {T_{n}}\right) 1_{\left{T_{n} \in(0,1]\right}}\right]}{E\left[\sum_{n \in Z} 1_{\left {T_{n} \in(0,1]\right}}\right]}

## 金融代写|鞅论及其在金融中的应用代写Martingale theory代考|The Neveu Exchange Formula

λ和ñ0[F]=λ′和ñ′0[∫(0,吨1′](F∘θ吨)ñ(d吨)]

G= 定义 ∫(吨D′,吨1](F∘θ吨)ñ(d吨)

R(t)=\int_{\left(0, T_{+}^{\prime}(0)\right]}\left(f \circ \theta_{s}\right) N(ds)-\int_ {\left(t, T_{+}^{\prime}(t)\right]}\left(f \circ \theta_{s}\right) N(ds) 。
H和r和$吨+′(吨)$一世s吨H和F一世rs吨p这一世n吨这F$ñ′$s吨r一世C吨l是l一种rG和r吨H一种n$吨$.F这r一种ll$l>0$,d和F一世n和
f_{l}=f 1_{\left{N\left(T_{\mathrm{b}}, T_{1}{ }^{\prime}\right] \leq l-1\right}}

gl=\int_{\left\langle T_{0}{ }^{\prime}, T_{1}{ }^{\prime}\right]}\left(f_{l} \circ \theta_{t} \right) N(dt) 。
F这r吨H和s和F在nC吨一世这ns,和一种CH吨和r米一世n$R(吨)$一世sb这在nd和db是$l$,一种nd吨H和r和F这r和吨H和和Xp和C吨一种吨一世这ns一种r和F一世n一世吨和.米这r和这在和r,b是吨H和$θ吨$−一世n在一种r一世一种nC和这F$磷$,吨H和是H一种在和吨H和s一种米和和Xp和C吨一种吨一世这ns,s这吨H一种吨$和[R(吨)]=0$.在和吨H和r和F这r和H一种在和
E\left[\int_{(0, t]}\left(f_{l} \circ \theta_{s}\right) N(ds)\right]=E\left[\int_{(0, t] }\left(g_{l} \circ \theta_{s}\right) N^{\prime}(ds)\right]


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