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# 金融代写|随机控制理论代写Stochastic Control代考|MA547 Stochastic s-dependence for self-similar sets

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## 金融代写|随机控制理论代写Stochastic Control代考|Stochastic s-dependence for self-similar sets

In this section some fractal sets are proven to be s-dependent showing that they do not satisfy condition 1s) of Definition 4.
Conditions under which the attractor of a finite family of similitudes and its boundary are s-dependent and two middle third Cantor sets are s-dependent are found. If coherent upper conditional probability is defined as in Theorem 5 we have that all these sets satisfy the factorization property, which is the standard definition of probabilistic dependence.

Let $\left(\Re^n, d\right)$ be the Euclidean metric space. A function $f: \Re^n \rightarrow \Re^n$ is called a contraction if $d(f(x), f(y)) \leq r d(x, y)$ for all $x, y \in \Re^n$, where $00$ but $h^s\left(f_i(E) \cap f_j(E)\right)=0$ for $i \neq j$ then $E$ is self similar. For any finite set of contractions there exists a unique non-empty compact invariant set $K$ (Falconer, 1986, Theorem 8.3), called attractor.
Given a finite set of contractions $\left{f_1, f_2, \ldots, f_m\right}$ we say that the open set condition (OSC) holds if there exists a bounded open set $O$ such that $O \subset \bigcup_{i=1}^m f_i(O)$ and $f_i(O) \cap f_j(O)=\oslash$ for $i \neq j$.

If $\left{f_1, f_2, \ldots, f_m\right}$ are similitudes with similarity ratios $r_i$ for $i=1 \ldots m$ the similarity dimension, which has the advantage of being easily calculable, is the unique positive number $s$ for which $\sum_{i=1}^m r_i^s=1$. If the OSC holds then the compact invariant set $K$ is self-similar and the Hausdorff dimension and the similarity dimension of $K$ are equal. If the similarity dimension is equal to $n$ then the interior of $K, K^0$ is non empty. In Lau (1999) it has been proven that, given a finite family of similitudes and the corresponding attractor $K$, if $K^0$ is non-void and Hausdorff dimension of $K$ is equal to $n$ then the Hausdorff dimension of the boundary of $K$ is less than $n$. Moreover since the lower density assumption holds for a self-similar set, from Proposition 1 of Section 4 we have that $K$ and its boundary are s-dependent. We can observe that if upper and lower probabilities are defined as in Theorem 5 then $K$ and its boundary satisfy the the factorization property.

## 金融代写|随机控制理论代写Stochastic Control代考|s-Dependence for Cantor sets

In the subjective probabilistic approach (de Finetti 1970, Dubins 1975 and Walley 1991) coherent upper conditional previsions $\bar{P}(\cdot \mid B)$ are functionals, defined on a linear space of bounded random variables, satisfying the axioms of coherence.

In Walley (1991) coherent upper conditional previsions are defined when the conditioning events are sets of a partition.

Definition 1. Let $\Omega$ be a non-empty set let $\boldsymbol{B}$ be a partition of $\Omega$. For every $B \in B$ let $\boldsymbol{K}(B)$ be a linear space of bounded random variables defined on $B$. Then separately coherent upper conditional previsions are functionals $\bar{P}(\cdot \mid B)$ defined on $K(B)$, such that the following conditions hold for every $X$ and $Y$ in $K(B)$ and every strictly positive constant $\lambda$ :

• 1) $\bar{P}(X \mid B) \leq \sup (X \mid B)$
• 2) $\bar{P}(\lambda X \mid B)=\lambda \bar{P}(X \mid B)$ (positive homogeneity);
• 3) $\bar{P}(X+Y) \mid B) \leq \bar{P}(X \mid B)+\bar{P}(Y \mid B)$;
• 4) $\bar{P}(B \mid B)=1$.

## 金融代写|随机控制理论代写STOCHASTIC CONTROL代考|SDEPENDENCE FOR CANTOR SETS

• 1) $\bar{P}(X \mid B) \leq \sup (X \mid B)$
• 2) $\bar{P}(\lambda X \mid B)=\lambda \bar{P}(X \mid B)$ positivehomogeneity;
• 3) $\bar{P}(X+Y) \mid B) \leq \bar{P}(X \mid B)+\bar{P}(Y \mid B)$
• 4) $\bar{P}(B \mid B)=1$.

## MATLAB代写

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