ABSTRACT
In this paper we consider numerical approximations of optimal stopping time problems for evolution equations in the general form
y'(t) =Mv(t)) y(0) =x [ ' }
posed in some Hilbert space H. The problem is to choose (for each initial state x) a stopping time for equation (1.1) so as to minimize the cost
J ( M ) = / e-Xsg(y(s))ds + *(y(t)). (1.2) Jo
over all t > 0. Here, the two terms play the roles respectively of a running and of a stopping cost. A numerical discretization of (1.1), (1.2) requires a finite dimensional (semi-discrete) approximation of (1.1), that is
and a corresponding discretization for (1.2), that is
Jn(PnX,t)= ( e-Xsg(yn(s))cis + <f>(yn(t)). (1.2n) Jo
We have shown in [Fe] that, under suitable assumptions, semi-discrete approximations of a large class of non-quadratic control problems provide minimizing sequences of approx imate optimal solutions. However, although the convergence theory is quite satisfactory, the application of Dynamic Programming techniques for the numerical computation of op timal solutions poses severe complexity problems. Our aim is precisely to use (1.1), (1.2) as a simple test problem to examine complexity and reliability of numerical schemes for its approximation.
