STOCHASTIC

378 MERTON
The model assumes that the individual "comes into" period t with wealth
invested in assets so that
W(t) = YJNi(,-h)Pi(t)- (7)
1
Notice that it is AW — h) because AW — h) is the number of shares
purchased for the portfolio in period (t — h) and it is Pit) because Pit)
is the current value of a share of the /-th asset. The amount of consumption
for the period, C(t)h, and the new portfolio, AW)> are simultaneously
chosen, and if it is assumed that all trades are made at (known) current
prices, then we have that
- C(t) h = t [AW) - AW - h)] Pit). (8)
I
The "dice" are rolled and a new set of prices is determined, P({t + h),
and the value of the portfolio is now £a AW) Pit + h). So the individual
"comes into" period (t + h) with wealth W(t + h) = ~Z" N^t) Pt(t + h)
and the process continues.
Incrementing (7) and (8) by h to eliminate backward differences, we
have that
and
-C(t + h) h = £ [N,(t + h)~ N,(t)] Pit + h)
I
= £ [Nt(t + h)- NitWlt + h) - Pit)}
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+ £ [AW + *) - AW) W) (9)
I
W(t + h) = YJ Nit) Pit + h). (10)
I
Taking the limits as h -* 0, 9 we arrive at the continuous version of (9)
and (10),
-C(t) dt = t dNlt) dPlt) + £ dNlt) Pit) (9')
1 1
9 We use here the result that Ito Processes are right-continuous [9, p. 15J and hence
Piit) and Wit) are right-continuous. It is assumed that C(t) is a right-continuous
function, and, throughout the papei, the choice of C(t) is restricted to this class of
functions.
626 PART V. DYNAMIC MODELS