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166 Risk management technology in financial services
9.3 Uncertainty modelling and risk control
Uncertainty, imprecision and vagueness are proactively present in all cases involving
risk measurement and control. The concepts discussed in section 9.2 characterize a
large number of exposures. The challenge is that of enriching these concepts with
mathematical operators as well as with numerical information.
Which are the problems associated with the use of probabilities in uncertainty
modelling?
One of them is that covering the whole landscape of a possibilistic function requires
an exhaustive list of mutually exclusive alternatives. Another constraint is that prob-
ability theory does not provide for representation of partial ignorance, or even repre-
sentation of total ignorance. This is important because in practically every case, given
the fact of inconsistency, ignorance produces information.
Moreover, the use of probability theory requires precise numbers, which do not
exist in an environment of uncertainty which characterizes risk management. A
Bayesian approach is more flexible and permits abduction (A given B). Its downside is
that of being error sensitive: errors can propagate through the system. As framework
for uncertainty modeling Boolean algebra uses statements of events. If k is an event,
If k is true then gk = 1 (9.1)
If k is false then gk =0 (9.2)
We don’t know if k is true. The function gk is member of a universe which
includes the values 0, 1,
3
but it is not probabilistic. Two approaches permit to avoid
the classical probability type restrictions. One is to use upper and lower probability
systems.
PA =P
1
A P
1
A (9.3)
where P
1
A is the lower bound and P
1
A the upper bound. If A
stands for not-A,
then: P
1
A+P
1
A
<1, but also P
1
A+P
1
A
>1, which looks like a contradiction,
but is acceptable, in pure probability theory. It is:
P
1
A =1 P
1
A
(9.4)
The approach explained in the preceding paragraph provides upper and lower
envelopes as well as monotone capacities. It also leads to belief functions within
the upper and lower envelopes. Within these limits we might work with probability
functions.
Total ignorance is expressed by means of the lower bound of probability: P
A =0.
This means that one is not certain at all. By contrast, if the upper bound is equal to
1, something is absolutely possible: P
1 =1.
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Using knowledge engineering for risk control 167
Possibility and belief (or necessity) measures can also lead to upper and lower
envelopes. The alternative, and better, way is to use straight possibility theory. If 
stands for a possibility function, then the measures characterizing possibility theory
can be expressed in the axioms:
4
0 =0 (9.5)
 =1 (9.6)
A +B = maxAB (9.7)
where stands for the universe of all value, whose possibility is equal to 1. The
possibility of zero is 0. Another basic axiom of possibility theory is:
MaxA A
 =1 (9.8)
Meaning that one of A or A
is possible. By contrast in probability theory:
PA +PA
=1 (9.9)
In possibility theory, the case of ignorance about 1 will be represented by:
A =A
=1 (9.10)
The meaning of equation (9.10) is that A is possible and not possible at the same
time.
These examples of possibilistic axioms point to the fact that the modelling of
uncertainty becomes feasible. Additionally, function gives much more freedom in
expressing real life events and decision procedure, doing so with less knowledge than
required by probabilistic functions.
The literature of possibility theory also uses functions of necessity, NA, which
practically stand for degree of certainty. An advantage of necessity functions is that
they are apodictic, an Aristotelian term meaning that a function is able to demonstrate
that something happens. A necessity measure is:
NA =1 A
(9.11)
Equation (9.11) means that if A is necessary then A
is impossible. The degree
of possibility is always greater than the degree of certainty or necessity. This is
expressed by:
NA A (9.12)
The modelling of uncertainty through possibility, necessity and belief functions
permits an accurate enough description of processes which are factual or judgmental,
like those characterizing the management of exposure. Each risk factor (see Chapter 2)

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