**5. Upper envelope**

8 Stochastic Control

*Xdhs* <sup>=</sup> <sup>1</sup>

The unconditional upper prevision is obtained as a particular case when the conditioning

Coherent upper conditional probabilities are obtained when only 0-1 valued random variables

**Theorem 2.** *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 m be a 0-1 valued finitely additive, but not countably additive, probability on* ℘(*B*) *such that a different m is chosen for each B. Thus, for each B* ∈ *B, the function*

*<sup>h</sup>s*(*B*) if 0 <sup>&</sup>lt; hs

Let *B* be a set with positive and finite Hausdorff outer measure in its Hausdorff dimension *s*. Denote by *<sup>h</sup><sup>s</sup>* the *<sup>s</sup>*-dimensional Hausdorff outer measure and for every *<sup>A</sup>* <sup>∈</sup> <sup>℘</sup>(*B*) by *<sup>μ</sup>*<sup>∗</sup>

have that the upper conditional prevision *P*(·|*B*) is a functional defined on **L**(*B*) with values

**Example 1** Let *B* = [0, 1]. The Hausdorff dimension of *B* is 1 and the Hausdorff 1-dimensional measure *h*<sup>1</sup> is the Lebesgue measure. Moreover *h*1[0, 1] = 1 then the coherent upper

We recall the definition of the Cantor set. Let *E*<sup>0</sup> = [0,1], *E*<sup>1</sup> = [0,1/3] ∪ [2/3,1], *E*<sup>2</sup> = [0,1/9] ∪ [2/9, 1/3] ∪ [2/3,7/9] ∪ [8/9,1], etc., where *En* is obtained by removing the open middle third

**Example 2** Let *B* be the Cantor set. The Hausdorff dimension of the Cantor set is *s* = *ln*<sup>2</sup>

*hs*(*B*) 

*<sup>h</sup>s*(*B*) = 1. Then the coherent upper conditional prevision is defined for every *<sup>X</sup>* <sup>∈</sup> **<sup>K</sup>**(*B*) by

*hs*(*B*) the upper conditional probability defined on ℘(*B*). From Theorem 1 we

*hs*(*B*) is a normalizing constant.

*<sup>B</sup> Xdh<sup>s</sup>* <sup>=</sup>

*<sup>B</sup> Xdh<sup>s</sup>* <sup>=</sup>

*<sup>B</sup> Xdh*<sup>1</sup>

*<sup>B</sup> Xdh ln*<sup>2</sup> *ln*3

(B) < +∞

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

*<sup>B</sup>*(*A*) =

<sup>3</sup>*<sup>n</sup>* . The Cantor's set

*ln*<sup>3</sup> and

(B) = 0, +∞

*hs*(*B*)

 *B Xdhs*.

1 *hs*(*B*)

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

*<sup>P</sup>*(*A*|*B*) = *<sup>m</sup>*(*AB*) if h<sup>s</sup>

event is Ω, that is *P*(*A*) = *P*(*A*|Ω) and *P*(*A*) = *P*(*A*|Ω).

are considered; they have been defined in [6] :

*is a coherent upper conditional probability.*

in � and the upper conditional probability *μ*<sup>∗</sup>

*Xdhs*. The number <sup>1</sup>

conditional prevision is defined for every *X* ∈ **K**(*B*) by

*<sup>n</sup>*=<sup>0</sup> *En*.

*<sup>P</sup>*(*X*|*B*) = <sup>1</sup>

of each interval in *En*−1, so *En* is the union of 2*<sup>n</sup>* intervals, each of length <sup>1</sup>

*<sup>P</sup>*(*X*|*B*) = <sup>1</sup>

*hs*(*B*) 

*defined on* ℘(*B*) *by*

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

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

is the perfect set *E* = <sup>∞</sup>

**4. Examples**

*and by*

*Xdμ*<sup>∗</sup>

 *B*

The last equality holds since each *B* is *hs*-measurable, that is *hs*(*B*) = *hs*(*B*).

A necessary and sufficient condition for an upper prevision *P* to be coherent is to be the *upper envelope* of linear previsions, i.e. there is a class *M* of linear previsions such that *P* =sup{*P*(*X*) : *P* ∈ *M*} [19, 3.3.3].

Given a coherent upper prevision *P* defined on a domain **K** the maximal coherent extension of *P* to the class of all bounded random variables is called [19, 3.1.1] *natural extension* of *P*.

The linear extension theorem [19, 3.4.2] assures that the class of all linear extensions to the class of all bounded random variables of a linear prevision *P* defined on a linear space **K** is the class *M*(*P*) of all linear previsions that are dominated by *P* on **K**. Moreover the upper and lower envelopes of *M*(*P*) are the natural extensions of *P* [19, Corollary 3.4.3].

Let *P*(·|*B*) be the coherent upper conditional prevision on the the class of all bounded Borel-measurable random variables defined on *B* defined in Theorem 1.

In Doria [9, Theorem 5] it is proven that, for every conditioning event *B*, the given upper conditional prevision is the upper envelope of all linear extensions of *P*(·|*B*) to the class of all bounded random variables on *B*.

**Theorem 3.** *Let* (Ω, *d*) *be a metric space and let B be a Borel-measurable partition of* Ω*. For every conditioning event B* ∈ *B let L*(*B*) *be the class of all bounded random variables defined on B and let P*(·|*B*) *be the coherent upper conditional previsionon the class of all bounded Borel-measurable random variables defined in Theorem 1. Then the coherent upper conditional prevision defined on L*(*B*) *as in Theorem 1 is the upper envelope of all linear extensions of P*(·|*B*) *to the class L*(*B*)*.*

In the same way it can be proven that the conjugate of the coherent upper conditional prevision *P*(·|*B*) is the *lower envelope* of *M*(*P*), the class of all linear extension of *P*(·|*B*) dominating *P*(·|*B*).
