Martingale Selection Theorem for a Stochastic Sequence with Relatively Open Convex Values

For a set-valued stochastic sequence (Gn)Nn=0 with relatively open convex values Gn(ω) we give a criterion for the existence of an adapted sequence (xn)Nn=0 of selectors, admitting an equivalent martingale measure. Mentioned criterion is expressed in terms of supports of the regular conditional upper distributions of the elements Gn. This result is a refinement of the main result of author’s previous paper (Teor. Veroyatnost. i Primen., 2005, 50:3, 480–500), where the sets Gn(ω) were assumed to be open and where were asked if the openness condition can be relaxed. Introduction Let (Ω,F ,P) be a probability space endowed with the filtration (Fn) N n=0. Consider a sequence of Fn-measurable set-valued maps Ω 7→ Gn(ω) ⊂ R , n = 0, . . . , N with the nonempty relatively open convex values Gn(ω). In this paper we give a criterion for the existence of a pair, consisting of an adapted single-valued stochastic process x = (xn) N n=1, xn(ω) ∈ Gn(ω) and a probability measure Q equivalent to P such that x is a martingale under Q. Following [1], we say that the martingale selection problem (m.s.p.) is solvable if such a pair (x,Q) exists. This problem is motivated by some questions of arbitrage theory. In particular, if the mappings Gn are single-valued, then we obtain the well-known problem concerning the existence of an equivalent martingale measure for a given stochastic process Gn = xn. In this case the solvability of the m.s.p. is equivalent to the absence of arbitrage in the market, where the discounted asset price process is described by x [2–5]. It is shown in [4] that an equivalent martingale measure for x exists iff the convex hulls of the supports of xn − xn−1 regular conditional distributions with respect to Fn−1 contain the origin as a point of relative interior [4, Theorem 3, condition (g)]. The aim of the present paper is to refine this result. In the framework of market models with transaction costs [6–8] the role of equivalent martingale measures is played by strictly consistent price processes. This name is assigned to P-martingales a.s. taking values in the relative interior of the random cones K, conjugate to the solvency cones K. Using the invariance of K under multiplication, it is easy to show (see [1, Introduction]) that the existence of a strictly consistent price process is 1991 Mathematics Subject Classification. 60G42.