Mathematical Modeling in Finance with Stochastic Processes

3.2 Ruin Probabilities

Section Starter Question

What is the solution of the equation xn = axn− 1 where ais a constant?
What kind of a function is the solution? What more, if anything, needs to
be known to obtain a complete solution?

Key Concepts

1. The probabilities, interpretation, meaning, and consequences of the
   “gambler’s ruin”.

Vocabulary

1. Classical Ruin Problem“Consider the familiar gambler who wins
   or loses a dollar with probabilitiespandq= 1−p, respectively play-
   ing against an infinitely rich adversary who is always willing to play
   although the gambler has the privilege of stopping at his pleasure. The
   gambler adopts the strategy of playing until he either loses his capital
   (“is ruined”) or increases it toa(with a net gain ofa−T 0 .) We are

3.3 Duration of the Gambler’s Ruin

distribution of the duration of the game. This is theclassical ruin

problem.”. (From W. Feller, inIntroduction to Probability Theory and

Applications, Volume I, Chapter XIV, page 342. [15])

Mathematical Ideas

Understanding a Stochastic Process

We consider a sequence of Bernoulli random variables,Y 1 ,Y 2 ,Y 3 ,...where
Yi= +1 with probabilitypandYi=−1 with probabilityq. We start with
an initial valueT 0. We define the sequence of sums Tn =

∑n
i=0Yi. We
are interested in the stochastic processT 1 ,T 2 ,T 3 ,.... It turns out this is a
complicated sequence to understand in full, so we single out particular simpler
features to understand first. For example, we can look at the probability that
the process will achieve the value 0 before it achieves the valuea. This is
a special case of a larger class of probability problems calledfirst-passage
probabilities.