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.

Minor clarifying changes; 37 pages

References in corpus (7)

• Arbitrage and duality in nondominated discrete-time models
• A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options
• Optimal Transport and Skorokhod Embedding
• Stability results for martingale representations: the general case
• Compactness in Adapted Weak Topologies
• Standardness of monotonic Markov filtrations
• Existence, Duality, and Cyclical monotonicity for weak transport costs

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• Regularity and stability of feedback relaxed controls
• Adapted Wasserstein distance between the laws of SDEs
• Extended Weak Convergence and Utility Maximization with Proportional Transaction Costs
• Bicausal optimal transport for SDEs with irregular coefficients
• Distributionally robust risk evaluation with a causality constraint and structural information
• The role of information in high dimensional stochastic optimal control