**7. Conclusions**

10 Stochastic Control

In [9, Theorem 4] it is proven that, if the conditioning event has positive and finite Hausdorff outer measure in its dimension *s* and **K**(*B*) is a linear lattice of bounded random variables defined on *B*, necessary conditions for the functional *P*(*X*|*B*) to be represented as Choquet

that *P*(*X*|*B*) is monotone, comonotonically additive, submodular and continuous from below. **Theorem 4.** *Let* (Ω, *d*) *be a metric space and let B be a Borel-measurable partition of* Ω*. For every <sup>B</sup>* <sup>∈</sup> *<sup>B</sup> denote by s the Hausdorff dimension of the conditioning event B and by h<sup>s</sup> the Hausdorff s-dimensional outer measure. Let K*(*B*) *be a linear lattice of bounded random variables defined on B. If the conditioning event B has positive and finite Hausdorff s-dimensional outer measure then the*

Moreover if the conditioning event *B* has positive and finite Hausdorff *s*-dimensional outer measure, from the properties of the Choquet integral ([5, Proposition 5.1]) the coherent upper

So the functional *P*(·|*B*) can be used to defined a *coherent risk measure* [1]. since it is monotone,

In [9, Theorem 6] sufficient conditions are given for a coherent upper conditional prevision to be uniquely represented as Choquet intergral with respect to its associated Hausdorff outer

**Theorem 5.** *Let* (Ω, *d*) *be a metric space and let B be a Borel-measurable partition of* Ω*. For every B* <sup>∈</sup> *<sup>B</sup> denote by s the Hausdorff dimension of the conditioning event B and by h<sup>s</sup> the Hausdorff s-dimensional outer measure. Let K*(*B*) *be a linear lattice of bounded random variables on B containing all constants. If B has positive and finite Hausdorff outer measure in its dimension and the coherent upper conditional prevision P*(·|*B*) *on K*(*B*) *is monotone, comonotonically additive, submodular and continuous from below then P*(·|*B*) *is representable as Choquet integral with respect to a monotone, submodular set function which is continuous from below. Furthermore all monotone set functions on* ℘(*B*) *with these properties agree on the set system of weak upper level sets*

*A* ∈ ℘(*B*)*. Let β be a monotone set function on* ℘(*B*)*, which is submodular, continuous from below*

An example is given in the particular case where **K**(*B*) is the linear space of all bounded Borel-measurable random variables on *B* and the restriction of the Hausdorff *s*-dimensional

*<sup>B</sup> Xdμ*<sup>∗</sup>

*<sup>B</sup>* <sup>=</sup> <sup>1</sup> *hs*(*B*) *<sup>B</sup> Xdhs.*

*<sup>B</sup>*, i.e. *<sup>P</sup>*(*X*|*B*) = <sup>1</sup>

*hs*(*B*)

*<sup>B</sup>*(*A*) = *<sup>h</sup><sup>s</sup>*(*AB*)

*hs*(*B*) *for*

*Xdhs*, are

integral with respect to the upper conditional probability *μ*∗

*i) monotone;*

*iii) submodular;*

measure.

*ii) comonotonically additive;*

*iv) continuous from below.*

v) translation invariant; vi) positively homogeneous;

conditional prevision *P*(·|*B*) is

*coherent upper conditional prevision P*(·|*B*) *defined on K*(*B*) *as in Theorem 2 is:*

subadditive, translation invariant and positively homogeneous.

*M* = {{*X* ≥ *x*} |*X* ∈ *K*(*B*), *x* ∈ �} *with the upper conditional probability μ*<sup>∗</sup>

*<sup>P</sup>*(*X*|*B*) =

outer measure to the Borel *σ*-field of subsets of *B* is considered.

*and such that represents P*(·|*B*) *as Choquet integral. Then the following equalities hold*

*<sup>B</sup> Xd<sup>β</sup>* <sup>=</sup>

In this chapter a model of coherent upper conditional precision is introduced. It is defined by the Choquet integral with respect to the *s*-dimensional Hausdorff outer measure if the conditioning event has positive and finite Hausdorff outer measure in its Hausdorff dimension *s*. Otherwise if the conditioning event has Hausdorff outer measure in its Hausdorff dimension equal to zero or infinity it is defined by a 0-1 valued finitely, but not countably, additive probability. If the conditioning event has positive and finite Hausdorff outer measure in its Hausdorff dimension the given upper conditional prevision, defined on a linear lattice of bounded random variables which contains all constants, is uniquely represented as the Choquet integral with respect Hausdorff outer measure if and only if it is a functional which is monotone, submodular, comonotonically additive and continuous from below.

Coherent upper conditional prevision based on the Hausdorff *s*-dimensional measure permits to analyze complex systems where information represented by sets with Hausdorff dimension less than *s*, have no influence on the situation; information represented by sets with the same Hausdorff dimension of the conditioning event can influence the system.

Coherent upper previsions defined by Hausdorff outer measures can also be applied in decision theory, to asses preferences between random variables defined on fractal sets and to defined coherent risk measures.
