STOCHASTIC

390 MERTON
Without loss of generality, assume that there are two assets, one "riskfree"
asset with return r and the other, a "risky" asset whose price is
log-normally distributed satisfying (40). From (28), the optimality equation
for J is
V
0 = " ~y y)1 e ~" \ I ~J L \ vl +J, +1(1 ~ y) ^
Jww
2-H[»W + -|2-(i-««*-")] &,
C*(/) = 7 ^ ^ (48)
8(>-exp[i^l(,-r)]) *
H'*W »X0 = ^ A WO + : ^ A (1 - e r "- r> ). (49)
20 It is assumed for simplicity that the individual has a zero bequest function, i.e.,
B = 0. If B(W, T) = H(T)(QW + by, the basic functional form for J in (47) will be
the same. Otherwise, systematic effects of age will be involved in the solution.
31 By Theorem I, there is no need to be concerned with uniqueness although, in this
case, the solution is unique.
638 PART V. DYNAMIC MODELS