STOCHASTIC

118 HOWARD M. TAYLOR
option which appears unfavorable under the assumption n = 0, cannot be favorable
for the investor if u > 0.
In the absolute random walk model where the price changes Zi , Z2, • • • are
normally distributed with mean y. > 0 and variance a 2 , one may show that the
sequence Oo = po £ ai £ (h £ • • • are bounded below, hence converge to a limit
denoted by a, that the functions Ho( •), i?i( •), • • • converge monotonically to
a limit function H(•), and that a = sup [y: H(y) = Oj. The recursion relationship
for Hj( •) in this case is Hj(y) = min {0, E{Hj-x(y + Z)\ + it], where Z
is normally distributed with mean n and variance a- 2 . This leads us to examine
the functional equation H{y) = min {0, E[H(y + Z)] + ji(. The solution to
this equation is not unique. One set of solutions is given by H*(y) = Ci +
exp {—2n(y + Cij/a 2 } — y, for any constants Ci and c2 such that Ci +
exp j— 2ji(y + C2)/o- 2 ) — y S 0 for all y. For, in this case, since
E[exp (-2itZ/cr 2 )] = 1 we have
E[H*(y + Z)] + M = C + exp j-2M(?/ +