Geometry of Distribution-Constrained Optimal Stopping Problems

A Dynamic Programming Principle for Distribution-Constrained Optimal Stopping

“…When the underlying filtration is general enough to allow for an independent uniformly distributed random variable, problem (1.1) is equivalent to its weak formulation where the supremum is also taken over potential probability spaces. This observation underlies the approach in Beiglböck et al [5]; specifically, identifying stopping times with measures on a certain canonical product space, the existence of an optimiser along with a monotonicity principle characterising the support set of any optimiser is obtained. In turn, for certain classes of cost functions, the latter is used to deduce the existence of so-called barrier-type solutions.…”

“…The problem is related to the so-called inverse first-passage-time problem which has a long history; it has also attracted recent attention: see e.g. [3,5,11], to which we refer for further motivation, references and an exposition of its role within financial and actuarial mathematics.…”

“…The DPP aside, we also study the stability of the distribution-constrained optimal stopping problem in that we establish continuity properties of the value of the problem as a function of the marginal constraint. First, following the arguments used to prove existence of an optimiser in [5], we obtain upper semi-continuity of this mapping under rather general assumptions on the cost function. Imposing additional regularity on the cost function, we also establish a certain 'right-continuity' of the value function; the result is of interest since it ensures that the value of the problem with a general constraint can be obtained as the limit of a sequence of approximating atomic problems which are easier to address by use of numerical methods.…”

“…For our approach it is crucial that the distribution-constrained problem (2.1) is in fact equivalent to the corresponding weak formulation. To formalise this, following [4] and [5], we will make use of the notion of randomised stopping times; see however also [13,16,17,18]. Let C 0 (R + ) denote the space of continuous functions from R + to R starting in zero, equipped with the topology of uniform convergence on compact sets; we denote the Wiener measure on C 0 (R + ) by W, and let the filtration F = (F t ) t≥0 be the usual augmentation of the canonical filtration.…”

“…Let C 0 (R + ) denote the space of continuous functions from R + to R starting in zero, equipped with the topology of uniform convergence on compact sets; we denote the Wiener measure on C 0 (R + ) by W, and let the filtration F = (F t ) t≥0 be the usual augmentation of the canonical filtration. Following [5], see also [4] to which we refer for further details, we then define the set RST(µ) of probability measures on the product space C 0 (R + ) × R + , referred to as randomised stopping times with pre-specified law 1 :…”