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# 金融代写|随机控制理论代写STOCHASTIC CONTROL代考|MATH691 Coherent conditional previsions and the Radon-Nikodym derivative

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## 金融代写|随机控制理论代写STOCHASTIC CONTROL代考|Coherent conditional previsions and the Radon-Nikodym derivative

Given a non-empty set $\Omega$ an outer measure is a function $\mu^{}: \wp(\Omega) \rightarrow[0,+\infty]$ such that $\mu^{}(\varnothing)=0, \mu^{}(A) \leq \mu^{}\left(A^{\prime}\right)$ if $A \subseteq A^{\prime}$ and $\mu^{}\left(\bigcup_{i=1}^{\infty} A_{i}\right) \leq \sum_{i=1}^{\infty} \mu^{}\left(A_{i}\right)$.

Examples of outer set functions or outer measures are the Hausdorff outer measures (Falconer 1986, Rogers 1998).

Let $(\Omega, d)$ be a metric space. A topology, called the metric topology, can be introduced into any metric space by defining the open sets of the space as the sets $G$ with the property:
if $x$ is a point of $G$, then for some $r>0$ all points y with $d(x, y)<r$ also belong to $G$.
It is easy to verify that the open sets defined in this way satisfy the standard axioms of the system of open sets belonging to a topology (Rogers, 1998, p.26).

The diameter of a non empty set $U$ of $\Omega$ is defined as $|U|=\sup {d(x, y): x, y \in U}$ and if a subset $A$ of $\Omega$ is such that $A \subset U_{i} U_{i}$ and $0<\left|U_{i}\right|<\delta$ for each i, the class $\left{U_{i}\right}$ is called a $\delta$-cover of $A$.

Let $\mathrm{s}$ be a non-negative number. For $\delta>0$ we define $h_{\mathrm{s}, \delta}(A)=\inf \sum_{i=1}^{+\infty}\left|U_{i}\right|^{5}$, where the infimum is over all $\delta$-covers $\left{U_{i}\right}$.
The Hausdorff s-dimensional outer measure of $A$, denoted by $h^{s}(A)$, is defined as
$$h^{s}(A)=\lim {\delta \rightarrow 0} h{s, \delta}(A) .$$

## 金融代写|随机控制理论代写STOCHASTIC CONTROL代考|The Choquet integral

We recall the definition of the Choquet integral (Denneberg, 1994) with the aim to define upper conditional previsions by Choquet integral with respect to Hausdorff outer measures and to prove their properties. The Choquet integral is an integral with respect to a monotone set function. Given a non-empty set $\Omega$ and denoted by $S$ a set system, containing the empty set and properly contained in $\wp(\Omega)$, the family of all subsets of $\Omega$, a monotone set function $\mu$ : $S \rightarrow \bar{\Re}{+}=\Re{+} \cup{+\infty}$ is such that $\mu(\oslash)=0$ and if $A, B \in S$ with $A \subseteq B$ then $\mu(A) \leq \mu(B)$.
Given a monotone set function $\mu$ on $S$, its outer set function is the set function $\mu^{}$ defined on the whole power set $\wp(\Omega)$ by $$\mu^{}(A)=\inf {\mu(B): B \supset A ; B \in S}, A \in \wp(\Omega)$$
The inner set function of $\mu$ is the set function $\mu_{}$ defined on the whole power set $\wp(\Omega)$ by $$\mu_{}(A)=\sup {\mu(B) \mid B \subset A ; B \in S}, A \in \wp(\Omega)$$
Let $\mu$ be a monotone set function defined on $S$ properly contained in $\wp(\Omega)$ and $X: \Omega \rightarrow \bar{\Re}=$ $\Re \cup{-\infty,+\infty}$ an arbitrary function on $\Omega$. Then the set function
$$G_{\mu, X}(x)=\mu{\omega \in \Omega: X(\omega)>x}$$
is decreasing and it is called decreasing distribution function of $X$ with respect to $\mu$. If $\mu$ is continuous from below then $G_{\mu, X}(x)$ is right continuous. In particular the decreasing distribution function of $X$ with respect to the Hausdorff outer measures is right continuous since these outer measures are continuous from below. A function $X: \Omega \rightarrow \bar{\Re}$ is called upper $\mu$ measurable if $G_{\mu^{}, X}(x)=G_{\mu_{}, X}(x)$. Given an upper $\mu$-measurable function $X: \Omega \rightarrow \bar{R}$ with decreasing distribution function $G_{\mu, X}(x)$, if $\mu(\Omega)<+\infty$, the asymmetric Choquet integral of $X$ with respect to $\mu$ is defined by
$$\int X d \mu=\int_{-\infty}^{0}\left(G_{\mu, \mathrm{X}}(x)-\mu(\Omega)\right) d x+\int_{0}^{\infty} G_{\mu, \mathrm{X}}(x) d x$$

## 全融代写|随机控制理论代写STOCHASTIC CONTROL代考|Coherent conditional previsions and the Radon-Nikodym derivative

\left 的分隔符缺失或无法识别
Hausdorff 的 $\mathrm{s}$ 维外测度 $A$ ，表示为 $h^{s}(A)$, 定义为
$\$ \$$\mathrm{h} \wedge{\mathrm{s}}(\mathrm{A})=\backslash \lim {\mid delta \mid rightarrow 0} h{\mathrm{~s}, \backslash delta }(\mathrm{A}). \ \$$

## 金融代写|随机控制理论代写STOCHASTIC CONTROL代考|The Choquet integral

$$\mu(A)=\inf \mu(B): B \supset A ; B \in S, A \in \wp(\Omega)$$

$$\mu(A)=\sup \mu(B) \mid B \subset A ; B \in S, A \in \wp(\Omega)$$

$$G_{\mu, X}(x)=\mu \omega \in \Omega: X(\omega)>x$$

$G_{\mu, X}(x)=G_{\mu, X}(x)$. 给定一个上 $\mu$ – 可测量函数 $X: \Omega \rightarrow \bar{R}$ 具有递咸分布函数 $G_{\mu, X}(x)$ ，如果 $\mu(\Omega)<+\infty$, 的不对称 Choquet 积分 $X$ 关于 $\mu$ 定义为
$$\int X d \mu=\int_{-\infty}^{0}\left(G_{\mu, \mathrm{X}}(x)-\mu(\Omega)\right) d x+\int_{0}^{\infty} G_{\mu, \mathrm{X}}(x) d x$$

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