STOCHASTIC

148 STEPHEN P. BRADLEY AND DWIGHT B. CRANE
subject to
-61* + *M(6I) + tf ,»(e,) = 0, Vei 6 Er,
— hi,i(ei) + *{,»(ei, e2) + A{,3(ei, e2) = 0, V(«i, e2) € #1 X E2,
bi" £ 0, «{,8(e,) S 0, tf .„(e,) £ 0, «{.,(«,, e„) fe 0, /h,3(ei, e») £ 0.
The subproblems for each security class that may be purchased in period two conditional
on the events of period one are V& and Vei 6 £1
Maximize
— fc2(«i) - 2/2*(ei) 23«2 T»(«I > ea)]6a*(ei)
(4.2) + J2'i [(1 + ?2.>(ei, e2))iri(ei, e2) + 02,a(ei, c2)ir4(ei, e2)]s2|3(ei, e2)
subject to
+ 53«, b(«i, ea)(j/2*(ei, e2) + »2,3(ei, e2))]M,s(ei, e2)
— &2*(ei) + s2,3(d , e2) + fc2.»(ei, e2) = 0, Ve2 € A'2,
t»2*(ei) 6 0, «2,,(e,, «,) S 0, /ij,j(e,, e2) £ 0.
The basic structure of the subproblems in each case is identical. When we consider
more than three time periods, we have the same structure only with necessary additional
variables and constraints. The underlying structure is simply that the hold
variables determine the amount of the commodity on hand at the start of the subsequent
period. This amount on hand is then either sold or held and the process repeated.
These subproblems turn out to be very easy to solve due to their simple structure
and homogeneity. First, every unbounded solution of subproblems of the form (4.1)
has hi* > 0. If not, the subproblem constraints imply the null solution. Similarly, every
unbounded solution of subproblems of the form (4.2) has 62*(ei) > 0. Further, if in
solving a subproblem it is profitable to buy one unit of a commodity it is profitable to
buy as much as possible, since the objective function is linear and the constraints are
homogeneous. Let us look at the problem of buying one unit of a particular commodity.
If it is profitable to do so we have constructed a ray yielding arbitrarily large profit for
an arbitrarily large multiple of that solution. Hence we have the following ray finding
problem which determines the selling and holding strategy that maximizes the return
from buying one unit of a security purchased in period one.
Maximize
subject to
£«! [(1 + fl r i,2(ei))ir2(ei) + ffi,2(ei) 21., ir4(ei , e2)]4.2(ei)
+ HH \yi k M £«i "^(ei. e2)]/ii.2(ei)
+ £.n, [(1 + ffi,a(ei, e2))ir3(ei, e2) + g\,i(ei , e2)jr4(ei, e2)]*?,J(e1, e2)
+ 2Z«i«i b( e i. 62) (2/1* + !>i,s(ei, e2))]fci,>(«i, e2)
si,2(ei) + A{,2(«i) = 1, Vei e A'i,
—Ai,s(ei) + «i,»(ei, et) + fti,»(«i, e2) = 0, V(ei, e2) € Ei X Et,
*J,2(«i) £ 0, tf,2(e,) £ 0, «{,,(«,, e2) 6 0, A{,3(ei, e2) 5; 0.
496 PART V. DYNAMIC MODELS