**2. Separately coherent upper conditional previsions**

Given a metric space (Ω, *d*) the Borel *σ*-field is the *σ*-field generated by the open sets of Ω. Let **B** be a Borel-measurable partition of Ω, i.e. all sets of the partition are Borel sets.

For every *B* ∈ **B** let us denote by *X*|*B* the restriction to *B* of a random variable defined on Ω and by sup(*X*|*B*) the supremum value that *X* assumes on *B*.

Separately coherent upper conditional previsions *P*(·|*B*) are functionals, defined on a linear space of bounded random variables, i.e. bounded real-valued functions, satisfying the axioms of separate coherence [19].

**Definiton 1.** Let (Ω, *d*) be a metric space and let **B** be a Borel-measurable partition of Ω. For every *B* ∈ **B** let **K**(*B*) be a linear space of bounded random variables on *B*. Separately coherent upper conditional previsions are functionals *P*(·|*B*) defined on **K**(B), such that the following conditions hold for every *X* and *Y* in **K**(*B*) and every strictly positive constant *λ*:


2 Stochastic Control

outer measure if the conditioning event has positive and finite Hausdorff outer measure in its dimension *s*. Otherwise if the conditioning event has Hausdorff outer measure in its dimension equal to zero or infinity it is defined by a 0-1 valued finitely, but not countably, additive probability. Coherent upper and lower conditional probabilities are obtained ([6])

If the conditioning event B has positive and finite Hausdorff outer measure in its Hausdorff dimension then the given upper conditional prevision defined on a linear lattice of bounded random variables is proven to be a functional, which is monotone, submodular, comonotonically additive and continuous from below. Moreover all these properties are proven to be a sufficient condition under which the upper conditional probability defined by Hausdorff outer measure is the unique monotone set function, which represent a coherent upper conditional prevision as Choquet integral. The given model of coherent upper

Many complex systems are strongly dependent on the initial conditions, that is small differences on the initial conditions lead the system to entirely different states. These systems are called *chaotic systems*. Thus uncertainty in the initial conditions produces uncertainty in the final state of the system. Often the final state of the system, called *strange attractor* is represented by a fractal set, i.e., a set with non-integer Hausdorff dimension. The model of coherent upper prevision, introduced in this chapter, can be proposed to forecast in a chaotic system when the conditional prevision of a random variable is conditioned to the attractor of

In Section 2 The notion of separately coherent conditional previsions and their properties are

In Section 3 separately coherent upper conditional previsions are defined in a metric space by the Choquet integral with respect to Hausdorff outer measure if the conditioning event has positive and finite Hausdorff outer measure in its dimension. Otherwise they are defined by

In Section 4 results are given such that a coherent upper conditional prevision, defined on a linear lattice of bounded random variables containing all constants, is uniquely represented as the Choquet integral with respect to its associated Hausdorff outer measure if and only if it

Given a metric space (Ω, *d*) the Borel *σ*-field is the *σ*-field generated by the open sets of Ω. Let

For every *B* ∈ **B** let us denote by *X*|*B* the restriction to *B* of a random variable defined on Ω

Separately coherent upper conditional previsions *P*(·|*B*) are functionals, defined on a linear space of bounded random variables, i.e. bounded real-valued functions, satisfying the axioms

**B** be a Borel-measurable partition of Ω, i.e. all sets of the partition are Borel sets.

conditional prevision can be applied to make prevision in chaotic systems.

when only 0-1 valued random variables are considered.

the chaotic system.

of separate coherence [19].

recalled.

The outline of the chapter is the following.

a 0-1 valued finitely, but not countably, additive probability.

**2. Separately coherent upper conditional previsions**

and by sup(*X*|*B*) the supremum value that *X* assumes on *B*.

is monotone, submodular and continuous from below.

Coherent upper conditional previsions can always be extended to coherent upper previsions on the class **L**(*B*) of all bounded random variables defined on *B*. If coherent upper conditional previsions are defined on the class **L**(*B*) no measurability condition is required for the sets *B* of the partition **B**.

Suppose that *P*(*X*|*B*) is a coherent upper conditional prevision on a linear space **K**(*B*) then its conjugate coherent lower conditional prevision is defined by *P*(*X*|*B*) = −*P*(−*X*|*B*). If for every *X* belonging to **K**(*B*) we have *P*(*X*|*B*) = *P*(*X*|*B*) = *P*(*X*|*B*) then *P*(*X*|*B*) is called a coherent *linear* conditional prevision (de Finetti [**?** ]) and it is a linear positive functional on **K**(*B*).

**Definition 2.** Let (Ω, *d*) be a metric space and let **B** be a Borel-measurable partition of Ω. For every *B* ∈ **B** let **K**(*B*) be a linear space of bounded random variables on *B*. Then linear coherent conditional previsions are functionals *P*(·|*B*) defined on **K**(*B*), such that the following conditions hold for every *X* and *Y* in **K**(*B*) and every strictly positive constant *λ*:

1') if *X* ≥ 0 then *P*(*X*|*B*) ≥ 0 (positivity); 2') *P*(*λX*|*B*) = *λP*(*X*|*B*) (positive homogeneity); 3') *P*(*X* + *Y*)|*B*) = *P*(*X*|*B*) + *P*(*Y*|*B*) (linearity); 4') *P*(*B*|*B*) = 1.

A class of bounded random variables is called a *lattice* if it is closed under point-wise maximum ∨ and point-wise minimum ∧.

Two random variables *X* and *Y* defined on *B* are *comonotonic* if, (*X*(*ω*1) − *X*(*ω*2))(*Y*(*ω*1) − *Y*(*ω*2)) ≥ 0 ∀*ω*1, *ω*<sup>2</sup> ∈ *B*.

**Definition 3.** Let (Ω, *d*) be a metric space and let **B** be a Borel-measurable partition of Ω. For every *B* ∈ **B** let **K**(*B*) be a linear lattice of bounded random variables defined on *B* and let *P*(·|*B*) be a coherent upper conditional prevision defined on **K**(*B*) then for every X, Y, *Xn* in **K**(B) *P*(·|*B*) is


A bounded random variable is called *B-measurable* or measurable with respect to the partition **B** [19, p.291] if it is constant on the atoms *B* of the partition. Let *G*(**B**) be the class of all **B**-measurable random variables.

The diameter of a non empty set *U* of Ω is defined as |*U*| = sup {*d*(*x*, *y*) : *x*, *y* ∈ *U*} and if a

*<sup>h</sup>s*(*A*) = lim*δ*→<sup>0</sup> *hs*,*δ*(*A*).

This limit exists, but may be infinite, since *hs*,*δ*(*A*) increases as *δ* decreases because less *δ*-covers are available. The *Hausdorff dimension* of a set *A*, *dimH*(*A*), is defined as the unique

(*A*)=+∞ if 0 ≤ s < dimH(A),

(*A*) = 0 if dimH(A) < s < +∞.

*<sup>h</sup>s*(*<sup>E</sup>* <sup>∪</sup> *<sup>F</sup>*) = *<sup>h</sup>s*(*E*) + *<sup>h</sup>s*(*F*) whenever *<sup>d</sup>*(*E*, *<sup>F</sup>*) = inf {*d*(*x*, *<sup>y</sup>*) : *<sup>x</sup>* <sup>∈</sup> *<sup>E</sup>*, *<sup>y</sup>* <sup>∈</sup> *<sup>F</sup>*} <sup>&</sup>gt; 0.

A subset *A* of Ω is called *measurable* with respect to the outer measure *h<sup>s</sup>* if it decomposes

*<sup>h</sup>s*(*E*) = *<sup>h</sup>s*(*<sup>A</sup>* <sup>∩</sup> *<sup>E</sup>*) + *<sup>h</sup>s*(*<sup>E</sup>* <sup>−</sup> *<sup>A</sup>*) for all sets *<sup>E</sup>* <sup>⊆</sup> <sup>Ω</sup>.

All Borel subsets of Ω are measurable with respect to any metric outer measure [11, Theorem 1.5]. So every Borel subset of Ω is measurable with respect to every Hausdorff outer measure

The restriction of *h<sup>s</sup>* to the *σ*-field of *hs*-measurable sets, containing the *σ*-field of the Borel sets, is called Hausdorff s-dimensional measure. In particular the Hausdorff 0-dimensional measure is the counting measure and the Hausdorff 1-dimensional measure is the Lebesgue

*<sup>h</sup>s*(*<sup>A</sup>* <sup>∪</sup> *<sup>B</sup>*) + *<sup>h</sup>s*(*<sup>A</sup>* <sup>∩</sup> *<sup>B</sup>*) = *<sup>h</sup>s*(*A*) + *<sup>h</sup>s*(*B*)

for every pair of Borelian sets *A* and *B*; so that [5, Proposition 2.4] the Hausdorff outer

*<sup>h</sup>s*(*<sup>A</sup>* <sup>∪</sup> *<sup>B</sup>*) + *<sup>h</sup>s*(*<sup>A</sup>* <sup>∩</sup> *<sup>B</sup>*) <sup>≤</sup> *<sup>h</sup>s*(*A*) + *<sup>h</sup>s*(*B*).

In [15, p.50] and [11, Theorem 1.6 (a)] it has been proven that if *A* is any subset of Ω there is a *Gσ*-set *G* containing *A* with *hs*(*A*) = *hs*(*G*). In particular *h<sup>s</sup>* is an *outer regular* measure.

Moreover Hausdorff outer measures are *continuous from below* [11, Lemma 1.3], that is for any

The Hausdorff *s*-dimensional measures are *modular* on the Borel *σ*-field, that is

We can observe that if 0 < *hs*(*A*) < +∞ then *dimH*(*A*) = *s*, but the converse is not true.

Let *<sup>s</sup>* be a non-negative number. For *<sup>δ</sup>* >0 we define *hs*,*<sup>δ</sup>* (*A*) = inf <sup>∑</sup>+<sup>∞</sup>

The *Hausdorff s-dimensional outer measure* of *A*, denoted by *hs*(*A*), is defined as

*hs*

*hs*

Hausdorff outer measures are *metric* outer measures:

every subset of Ω additively, that is if

*h<sup>s</sup>* since Hausdorff outer measures are metric.

increasing sequence of sets {*Ai*} we have

*<sup>i</sup> Ui* and 0 < |*Ui*| < *δ* for each i, the class {*Ui*} is called a

Hausdorff Outer Measures to Forecast in Chaotic Dynamical Systems

<sup>ß</sup>=<sup>1</sup> |*Ui*| *s*

Coherent Upper Conditional Previsions De ned by

, where the

55

subset *<sup>A</sup>* of <sup>Ω</sup> is such that *<sup>A</sup>* ⊂

infimum is over all *δ*-covers {*Ui*}.

*δ*-cover of *A*.

value, such that

measure.

measures are *submodular*

Denote by *P*(*X*|**B**) the random variable equal to *P*(*X*|*B*) if *ω* ∈ *B*.

Separately coherent upper conditional previsions *P*(*X*|*B*) can be extended to a common domain **H** so that the function *P*(·|**B**) can be defined from **H** to *G*(**B**) to summarize the collection of *P*(*X*|*B*) with *B* ∈ **B**.

*P*(·|**B**) is assumed to be separately coherent if all the *P*(·|*B*) are separately coherent. In Theorem 1 [9] the function *P*(*X*|**B**) is compared with the Radon-Nikodym derivative.

It is proven that, every time that the *σ*-field of the conditioning events is properly contained in the *σ*-field of the probability space and it contains all singletons, the Radon-Nikodym derivative cannot be used as a tool to define coherent conditional previsions. This is due to the fact that one of the defining properties of the Radon-Nikodym derivative, that is to be measurable with respect to the *σ*-field of the conditioning events, contradicts a necessary condition for the coherence.

Analysis done points out the necessity to introduce a different tool to define coherent conditional previsions.
