All adapted topologies are equal
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All adapted topologies are equal Julio Backhoff-Veraguas1 · Daniel Bartl1 · Mathias Beiglböck1
· Manu Eder1
Received: 9 May 2019 / Revised: 27 May 2020 / Published online: 14 September 2020 © The Author(s) 2020
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. Keywords Aldous’ extended weak topology · Hellwig’s information topology · Nested distance · Causal optimal transport · Stability of optimal stopping · Vershik’s iterated Kantorovich distance Mathematics Subject Classification Primary 60G42 · 60G44; Secondary 91G20 The first author acknowledges support through FWF-project P28661. The third and the fourth author acknowledge financial support through FWF-project Y00782..
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Mathias Beiglböck [email protected] Julio Backhoff-Veraguas [email protected] Daniel Bartl [email protected] Manu Eder [email protected]
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Department of Mathematics, University of Vienna, Vienna, Austria
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1 Introduction 1.1 Outline If some type of natural phenomenon is modelled through a stochastic process, one might expect that the model does not describe reality in an entirely accurate way. To be able to study the impact of such inaccuracies on the problems one is trying to solve, it makes sense to equip the set of laws of stochastic processes with a suitable notion of distance or topology. Denoting by := X N the path space (where X is some Polish space and N ∈ N), the set of laws of stochastic processes is P(), i.e. the set of probability measures on . Clearly, P() carries the usual weak topology. However, this topology does not respect the time evolution of stochastic processes which has a number of potentially inconvenient consequences: e.g., problems of optimal stopping/utility maximization/stochastic programming are not continuous, arbitrary processes can be approximated by processes which are dete
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