All adapted topologies are equal

Abstract

A number of researchers have introduced topological structures on the set of laws of stochastic processes. A unifying goal of these authors is to strengthen the usual weak topology in order to adequately capture the temporal structure of stochastic processes. Aldous defines an extended weak topology based on the weak convergence of prediction processes. In the economic literature, Hellwig introduced the information topology to study the stability of equilibrium problems. Bion–Nadal and Talay introduce a version of the Wasserstein distance between the laws of diffusion processes. Pflug and Pichler consider the nested distance (and the weak nested topology) to obtain continuity of stochastic multistage programming problems. These distances can be seen as a symmetrization of Lassalle’s causal transport problem, but there are also further natural ways to derive a topology from causal transport. Our main result is that all of these seemingly independent approaches define the same topology in finite discrete time. Moreover we show that this ‘weak adapted topology’ is characterized as the coarsest topology that guarantees continuity of optimal stopping problems for continuous bounded reward functions.

Robustness to Approximations and Model Learning in MDPs and POMDPs

“…weakly and the conditional kernels on the future random variables given the past converge weakly as well. This has been termed as extended weak convergence [1] or convergence under the the information topology [50]; these have recently shown to be equivalent in discrete-time [4,Theorem 1.1]. Applications of this notion of convergence to robustness to system models in stochastic control have recently been studied in different contexts [7,5].…”

Adapted Wasserstein distances and stability in mathematical finance

“…Yet another idea to account for the temporal evolution of processes would be to symmetrise the causal transport costs W c (P, Q) defined by Lassalle [41] by taking the maximum or sum of W 2 c (P, Q) and W 2 c (Q, P); this was pointed out by Soumik Pal. In parallel work [6], the four authors of the present article investigate the relations between these concepts in detail. Remarkably, in (finite) discrete time, all of the concepts mentioned above (adapted Wasserstein distances, extended weak convergence, information topology, nested distances, symmetrised causal transport costs) define the same topology.…”

“…As noted above, this "weak adapted topology" refines the usual weak topology (properly for T ≥ 2; see also Remark 5.2). The articles [8,6,23] investigate basic properties of this topology; e.g., the weak adapted topology is Polish [8,Sect. 5], and sets are totally bounded with respect to the adapted Wasserstein distance/nested distance if and only if they are totally bounded with respect to the usual Wasserstein distance [6,Lemma 1.6].…”

“…The articles [8,6,23] investigate basic properties of this topology; e.g., the weak adapted topology is Polish [8,Sect. 5], and sets are totally bounded with respect to the adapted Wasserstein distance/nested distance if and only if they are totally bounded with respect to the usual Wasserstein distance [6,Lemma 1.6]. For recent applications of these concepts to optimal transport and probabilistic variants thereof, we refer to Backhoff-Veraguas et al [11,12] and Wiesel [54].…”

Quantization and martingale couplings

“…Numerous financial applications of this adapted Wasserstein distance have been investigated by Backhoff-Veraguas et al (2020a). According to Backhoff-Veraguas et al (2020b), the topology induced by this distance is equal to the other adapted topologies which had been introduced in particular in view of financial applications.…”