Mathematical Modeling in Finance with Stochastic Processes

96 CHAPTER 3. FIRST STEP ANALYSIS FOR STOCHASTIC PROCESSES

Theorems about Ruin Probabilities

Consider a gambler who wins or loses a dollar on each turn of a game with
probabilitiesp andq = 1−prespectively. Let his initial capital be T 0.
The game continues until the gambler’s capital either is reduced to 0 or has
increased toa. LetqT 0 be the probability of the gambler’s ultimate ruin
andpT 0 the probability of his winning. We shall show later that (see also
Duration of the Game Until Ruin.)

pT 0 +qT 0 = 1

so that we need not consider the possibility of an unending game.

Theorem 1.The probability of the gambler’s ruin is

qT 0 =

(q/p)a−(q/p)T^0

(q/p)a− 1

ifp 6 =qand
qT 0 = 1−T 0 /a

ifp=q= 1/ 2.

Proof.After the first trial the gambler’s fortune is eitherT 0 −1 orT 0 + 1 and
therefore we must have

qT 0 =pqT 0 +1+qqT 0 − 1 (3.1)

provided 1< T 0 < a−1. ForT 0 = 1, the first trial may lead to ruin, and
(3.1) is replaced by
q 1 =pq 2 +q.

Similarly, forT 0 =a−1 the first trial may result in victory, and therefore

qa− 1 =qqa− 2.

To unify our equations, we define as a natural convention thatq 0 = 1, and
qa= 0. Then the probability of ruin satisfies (3.1) forT 0 = 1, 2 ,...,a−1.
This defines a set ofa−1 difference equations, with boundary conditions at
0 anda. If we solve the system of difference equations, then we will have the
desired probabilityqT 0 for any value ofT 0.