**1. Introduction**

Coherent conditional previsions and probabilities are tools to model and quantify uncertainties; they have been investigated in de Finetti [3], [4], Dubins [10] Regazzini [13], [14] and Williams [20]. Separately coherent upper and lower conditional previsions have been introduced in Walley [18], [19] and models of upper and lower conditional previsions have been analysed in Vicig et al. [17] and Miranda and Zaffalon [12].

In the subjective probabilistic approach coherent probability is defined on an arbitrary class of sets and any coherent probability can be extended to a larger domain. So in this framework no measurability condition is required for random variables. In the sequel, bounded random variables are bounded real-valued functions (these functions are called *gambles* in Walley [19] or *random quantities* in de Finetti [3]). When a measurability condition for a random variable is required, for example to define the Choquet integral, it is explicitly mentioned through the paper.

Separately coherent upper conditional previsions are functionals on a linear space of bounded random variables satisfying the axioms of separate coherence. They cannot always be defined as an extension of conditional expectation of measurable random variables defined by the Radon-Nikodym derivative, according to the axiomatic definition. It occurs because 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 coherence (see Doria [9, Theorem 1], Seidenfeld [16]).

So the necessity to find a new mathematical tool in order to define coherent upper conditional previsions arises.

In Doria [8], [9] a new model of coherent upper conditional prevision is proposed in a metric space. It is defined by the Choquet integral with respect to the *s*-dimensional Hausdorff

©2012 Doria, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Doria, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 2 Stochastic Control 52 Stochastic Modeling and Control Coherent Upper Conditional Previsions Defined by Hausdorff Outer Measures to Forecast in Chaotic Dynamical Systems <sup>3</sup>

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]) when only 0-1 valued random variables are considered.

**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

53

Coherent Upper Conditional Previsions De ned by

Hausdorff Outer Measures to Forecast in Chaotic Dynamical Systems

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*

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

**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*:

A class of bounded random variables is called a *lattice* if it is closed under point-wise

Two random variables *X* and *Y* defined on *B* are *comonotonic* if, (*X*(*ω*1) − *X*(*ω*2))(*Y*(*ω*1) −

**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

ii) *comonotonically additive* iff *P*(*X* + *Y*|*B*) = *P*(*X*|*B*) + *P*(*Y*|*B*) if *X* and *Y* are comonotonic;

iv) *continuous from below* iff *limn*→∞*P*(*Xn*|*B*) = *P*(*X*|*B*) if *Xn* is an increasing sequence of

conditions hold for every *X* and *Y* in **K**(*B*) and every strictly positive constant *λ*:

1) *P* (*X*|*B*) ≤ sup(*X*|*B*);

4) *P*(*B*|*B*) = 1.

of the partition **B**.

4') *P*(*B*|*B*) = 1.

**K**(B) *P*(·|*B*) is

*Y*(*ω*2)) ≥ 0 ∀*ω*1, *ω*<sup>2</sup> ∈ *B*.

**K**(*B*).

2) *P*(*λ X*|*B*) = *λ P*(*X*|*B*) (positive homogeneity); 3) *P*(*X* + *Y*)|*B*) ≤ *P*(*X*|*B*) + *P*(*Y*|*B*) (subadditivity);

1') if *X* ≥ 0 then *P*(*X*|*B*) ≥ 0 (positivity);

maximum ∨ and point-wise minimum ∧.

2') *P*(*λX*|*B*) = *λP*(*X*|*B*) (positive homogeneity); 3') *P*(*X* + *Y*)|*B*) = *P*(*X*|*B*) + *P*(*Y*|*B*) (linearity);

i) *monotone* iff *X* ≤ *Y* implies *P*(*X*|*B*) ≤ *P*(*Y*|*B*);

random variables converging to *X*.

iii) *submodular* iff *P*(*X* ∨ *Y*|*B*) + *P*(*X* ∧ *Y*|*B*) ≤ *P*(*X*|*B*) + *P*(*Y*|*B*);

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

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 the chaotic system.

The outline of the chapter is the following.

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

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

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 is monotone, submodular and continuous from below.
