STOCHASTIC

400 MERTON
To derive (91), an "artificial" state variable, x(t), is constructed with
x(t) = 0 while the individual is alive and x(t) — 1 in the event of death.
Therefore, the stochastic process which generates x is defined by
dx = dq and y ^ 1 with probability one (92)
and T is now defined by x as
T = min{f | t > 0 and x(t) = 1}. (93)
The derived utility function, J, can be considered a function of the state
variables W, x, and t subject to the boundary condition
J(W, x, t) = B(W, t) when x = 1. (94)
In this form, example three is shown to be of the same type as examples
one and two in that the occurrence of the Poisson event causes a state
variable to be incremented, and (91) is of the same form as (78) and (83).
A comparison of (91) for the particular case when B = 0 (no bequests)
with (82) suggested the following theorem. 31
THEOREM VI. If r is as defined in (93) and U is such that the integral
E0[f„ U(C, t) dt] is absolutely convergent, then the maximization of
E„[f0 U(C, t) dt] is equivalent to the maximization of