STOCHASTIC

CALL OPTIONS AND TIMING STRATEGY IN THE STOCK MARKET 119
tion may be written
R(y) = min \y, E[R(v + AF)]}
— y for y g a
= E[B(y + AY)] for y > a.
We expand in a Taylor series and take expectation to get:
and
R(y + AY) = R(y) + AY-R'(y) + h{AY?R" (y) + •••
E[R(y + AY)} = R(y) + y.R'(y) • At +(a 2 /2)R"(y) • At +
For the region }>«we thus arrive at the differential equation
0 = nR'iy) + (