3-Milestones Page 80 Wednesday, February 4, 2004 12:47 PM
80 The Mathematics of Financial Modeling and Investment Managementpected (i.e., a random disturbance). As a consequence, prices move as
martingales, as argued by Bachelier. Bachelier was, in fact, the first to
give a precise mathematical structure in continuous time to price pro-
cesses subject to competitive pressure by many agents.THE RUIN PROBLEM IN INSURANCE: LUNDBERG
In Uppsala, Sweden, in 1903, three years after Bachelier defended his
doctoral dissertation in Paris, Filip Lundberg defended a thesis that was
to become a milestone in actuarial mathematics: He was the first to
define a collective theory of risk and to apply a sophisticated probabilis-
tic formulation to the insurance ruin problem. The ruin problem of an
insurance company in a nonlife sector can be defined as follows. Sup-
pose that an insurance company receives a stream of sure payments
(premiums) and is subject to claims of random size that occur at random
times. What is the probability that the insurer will not be able to meet
its obligations (i.e., the probability of ruin)?
Lundberg solved the problem as a collective risk problem, pooling
together the risk of claims. To define collective risk processes, he intro-
duced marked Poisson processes. Marked Poisson processes are pro-
cesses where the random time between two events is exponentially
distributed. The magnitude of events is random with a distribution inde-
pendent of the time of the event. Based on this representation, Lundberg
computed an estimate of the probability of ruin.
Lundberg’s work anticipated many future developments of probability
theory, including what was later to be known as the theory of point pro-
cesses. In the 1930s, the Swedish mathematician and probabilist Harald
Cramer gave a rigorous mathematical formulation to Lundberg’s work. A
more comprehensive formal theory of insurance risk was later developed.
This theory now includes Cox processes—point processes more general
than Poisson processes—and fat-tailed distributions of claim size.
A strong connection between actuarial mathematics and asset pric-
ing theory has since been established.^6 In well-behaved, complete mar-
kets (see Chapter 23), establishing insurance premiums entails principles
that mirror asset prices. In the presence of complete markets, insurance
would be a risk-free business: There is always the possibility of reinsur-
ance. In markets that are not complete—essentially because they make
unpredictable jumps—hedging is not possible; risk can only be diversi-(^6) Paul Embrechts, Claudia Klüppelberg, and Thomas Mikosch, Modelling Extremal
Events for Insurance and Finance (Berlin: Springer, 1996).