OPTIMAL STOPPING OF CONTINUOUS TIME STOCHASTIC PROCESSES AND STOCHASTIC DIFFERENTIAL REPRESENTATIONS FOR THE VALUE FUNCTIONS M. A. Shashiashvili UDC 519.21 Introduction Throughout this paper we will assume given: I) a complete probability state (~,~,P); 2) a nondecreasing right-continuous family of ~-subalgebras (~,t~0) of the o-algebra ; all the ~t are assumed to be completed sets of P-measure zero; 3) a completely measurable stochastic process (x,,~,,t~0) such that Mx~< ~ for every te~, ~) 9 We put v, = sup Mx~, (1) where the supremum is taken over all stopping times (s.t.) x, for which P (t~<~)=l and Mx T is defined (i.e., is-not necessarily finite). Let x~=li--mxt (where x~ is a random variable, since the process x t is measurable and consequently the corollary of Choquet's theorem is applicable [I, Chap. I, Sec. 32]) and define for every Markov time ~ the random variable x,=l x,(w~, if w~{w:'~(w)< m }, 1 x~o, if we{w:x(w)= oo }. Together with (i) we consider vt = sup Mx~, (2) where the supremum is taken over all Markov times (m.t.) T for which P (t~x~ ~)=I and Mx~ is defined. We will say that the s.t. T e is (c, vt) optimal if v,-~ Mx=~ . Similarly, we say that the m.t. T C is (E, vt) optimal if ~,-~Mx~ (in the case when E equals zero, the time To will be called optimal). In Secs. 1-3 we give some generalizations of some of the main results in the papers [2, 3] concerning a supermartingale characterization of the values v t and V t and of the E giving optimal rules. In Secs. 4-5 we derive a stochastic differential representation for the minimal super- martingale majorizing a given stochastic process. The main theorem in the paper is Theorem 7 in Sec. 5. In Sec. 6 we discuss the problesn betweem optimal stopping and a '~ariational inequality," and in the last section, Sec. 7, we derive a "smooth fit" condition for Ito processes. The author expresses his deep thanks to A. N. Shiryaev and V. M. Dochvir' for their help in this work. Institute of Economics and Law, Academy of Sciences of the Georgian SSR. Litovskii Matematicheskii Sbornik (Lietuvos Matematiko Rinkinys), Vol. 19, No. January-March, 1979. Original article submitted August 19, 1977. Translated from i, pp. 197-211, 140 0363-1672/79/1901-0140507.50 9 1979 Plenum Publishing Corporation