Martingales on Random Sets and the Strong Martingale Property

Abstract

Abstract. Let X be a process defined on an optional random set. The paper develops two different conditions on X guaranteeing that it is the restriction of a uniformly integrable martingale. In each case, it is supposed that X is the restriction to Λ of some special semimartingale Z with canonical decomposition Z = M + A. The first condition, which is both necessary and sufficient, is an absolute continuity condition on A. Under additional hypotheses, the existence of a martingale extension can be characterized by a strong martingale property of X on Λ. Uniqueness of the extension is also considered. IntroductionLet Λ be an optional random set and let X t (ω) be defined for (t, ω) ∈ Λ. We consider the following and some of its extensions. (0.1) Problem. Find necessary and sufficient conditions on X guaranteeing that it is the restriction to Λ of a globally defined, right continuous uniformly integrable martingale.For an example where this formulation may be natural, consider a process (Y t ) t≥0 with values in a manifold. Given a coordinate patch V , let Λ := {(t, ω) : Y t (ω) ∈ V } and let X t (ω) denote a real component of Y t (ω) for (t, ω) ∈ Λ. A second natural example is provided by X = f •W , where W is a Markov process in a state space E and f is a function defined on a subset S of E, Λ denoting in this case {(t, ω) :The solution is obvious if there is an increasing sequence of stopping times T n which are complete sections of Λ (i.e., T n ⊂ Λ and P{T n < ∞} = 1) such that Λ ⊂ ∪ n 0, T n . A number of other cases may now be found in the literature. In sections 2 and 3, we give a complete solution for the discrete parameter problem under mild conditions on X. The continuous parameter case, treated in sections 4 and 5, involves considerable additional complication. Roughly speaking, in the discrete parameter case the condition is that X have a "strong martingale property" on Λ (defined in (3.4)), but in the continuous parameter context, an example is given in section 5 to show that this condition is not sufficient. Theorem (4.1), one of the main results of the paper, assumes X is the restriction of a special semimartingale Z = M + A (with A predictable and of integrable 1991 Mathematics Subject Classification. 60J30.

On the Loss of the Semimartingale Property at the Hitting Time of a Level

“…We stress that Question II appears not to have been treated in these papers. Finally, we discuss (omitting certain technical details) the relations between our treatment of Question II and the work in [16]. In [16] a process X on an optional random set Λ is considered and the question of interest is whether X is a restriction to Λ of a globally defined martingale (this question arises naturally in the setting of semimartingales on manifolds, when a semimartingale defined on the entire manifold satisfies the martingale property on each chart).…”

“…The analysis in [16] is performed under the standing assumption that X is the restriction to Λ of some special semimartingale. Hence, our Question II is precisely the question of whether this standing assumption holds.…”

“…In this paper we give explicit deterministic if-and-only-if conditions in the diffusion setting for this assumption to be satisfied in the case the optional set is of the form Λ = [0, ζ). We should, however, note that the study in [16] is particularly interesting when Λ is non-predictable. Thus, the present paper and [16], in fact, study distinct questions tailored to different settings.…”

“…At this juncture we refer to [10], [6, Sec. V.1], [17], [19], [15], and [16], where several classes of processes on stochastic intervals (or even on optional random sets) are considered. In particular, in [10] (also see [14, Ch.…”

“…We should, however, note that the study in [16] is particularly interesting when Λ is non-predictable. Thus, the present paper and [16], in fact, study distinct questions tailored to different settings.…”