STOCHASTIC

This gives
with
R. G. VICKSON
t(t) = f ' % + Tt + MyW+W'0 - F W> ')]• W
III. Stochastic Optimal Control
A typical stochastic optimal control problem for an Ito process is
max£ f{X{t),v,t) dt, (xi)
v J0
dX = f(X,v,t)dt + g{X,v,t)dz, (xii)
where v is a "control" function, constrained to be in some infinite-dimensional
set V. To obtain optimality conditions for this problem, it is convenient to
replace it by an approximating discrete time problem:
N
max E £ 5 • f(Xt, vt, tt) (xiii)
s.t.
5Xt = f(Xhv„td-S + g(X„v„td • zt{5), (xiv)
and veV. Using the principle of optimality, the optimal control vt at stage
i, given Xt = xt, is the solution to
Jt (xt, ti) = max [