New characterizations of the $S$ topology on the Skorokhod space

Abstract

The S topology on the Skorokhod space was introduced by the author in 1997 and since then it has proved to be a useful tool in several areas of the theory of stochastic processes. The paper brings complementary information on the S topology. It is shown that the convergence of sequences in the S topology admits a closed form description, exhibiting the locally convex character of the S topology. Morover, it is proved that the S topology is, up to some technicalities, finer than any linear topology which is coarser than Skorokhod's J 1 topology. The paper contains also definitions of extensions of the S topology to the Skorokhod space of functions defined on [0, +∞) and with multidimensional values.

Martingale optimal transport duality

“…Let Ω be a non-empty subset of the Skorokhod space D([0, T ]; R d + ) of all càdlàg functions ω : [0, T ] → R d + that is closed with respect to Jakubowski's S-topology [39,40]. We denote the relative topology of S on Ω again by S and, similarly to [46], endow Ω with the coarsest topology S * making all S-continuous functions ξ : Ω → R continuous.…”

“…This definition of a topology is known as the Kantorovich-Kisyński recipe; see [47] or [30, Sections 1.7.18, 1.7.19 on pages 63-64]. In particular, it is discussed in [40,Appendix] that {ν n } n∈N converges to ν * in the (a posteriori) S-topology, if every subsequence {ν n k } k∈N has a further subsequence {ν n k l } l∈N such that ν n k l ⇀ S ν * .…”

“…As a different example, if one starts with almost-sure convergence as the a priori convergence (instead of the ⇀ S convergence as above), then the resulting a posteriori convergence is the convergence-in-probability; see [40].…”

“…The following fact from [39,40] is an essential ingredient of our continuity proof. Recall the up-crossings U a,b,i t of Definition 6.2.…”

“…In this paper, we study general martingale optimal transport on a subset Ω of the Skorokhod space D([0, T ]; R d + ) of all R d + -valued càdlàg functions, i.e., functions ω : [0, T ] → R d + that are continuous from the right and have finite left limits. We assume that Ω is a closed subset of D([0, T ]; R d + ) with respect to Jakubowski's S-topology [39,40] and endow it with a regularized version of S. Our main goal is to prove duality with the same Choquet capacity defined by countably additive (martingale) measures, for different choices of X by appropriately extending the quotient set.…”