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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,

gk provides a grade of confidence about the truth of k.

If k is true then gk = 1 (9.1)

If k is false then gk =0 (9.2)

We don’t know if k is true. The function gk 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.

PA =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 = maxAB (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:

MaxA A

=1 (9.8)

Meaning that one of A or A

is possible. By contrast in probability theory:

PA +PA

=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, NA, 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:

NA =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:

NA ≤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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