Mathematical Modeling in Finance with Stochastic Processes

3.2. RUIN PROBABILITIES 97

Note that we can rewrite the difference equations as

pqT 0 +qqT 0 =pqT 0 +1+qqT 0 − 1.

Then we can rearrange and factor to obtain

qT 0 +1−qT 0

qT 0 −qT 0 − 1

=

q

p

This says the ratio of successive differences ofqT 0 is constant. This suggests
thatqT 0 is a power function,
qT 0 =λT^0

since power functions have this property.
We first take the case whenp 6 =q. Then based on the guess above (or
also on standard theory for linear difference equations), we try a solution of
the formqT 0 =λT^0. That is

λT^0 =pλT^0 +1+qλT^0 −^1.

This reduces to
pλ^2 −λ+q= 0.

Sincep+q= 1, this factors as

(pλ−q)(λ−1) = 0,

so the solutions areλ=q/p, andλ= 1. (One could also use the quadratic
formula to obtain the same values, of course.) Again by the standard theory
of linear difference equations, the general solution is

qT 0 =A·1 +B·(q/p)T^0 (3.2)

for some constantsA, andB.
Now we determine the constants by using the boundary conditions:

q 0 =A+B= 1

qa=A+B(q/p)a= 0.

Solving, substituting, and simplifying:

qT 0 =

(q/p)a−(q/p)T^0

(q/p)a− 1

.