Abstract
Two adapted stochastic processes can have similar laws but give different results in applications such as optimal stopping, queuing theory, or stochastic programming. The reason is that the topology of weak convergence does not account for the growth of information over time that is captured in the filtration of an adapted stochastic process. To address such discontinuities, Aldous introduced the extended weak topology, and subsequently, Hoover and Keisler showed that both, weak topology and extended weak topology, are just the first two topologies in a sequence of topologies that get increasingly finer. We introduce higher rank expected signatures to embed adapted processes into graded linear spaces and show that these embeddings induce the adapted topologies of Hoover–Keisler.
Funding Statement
PB is supported by the Engineering and Physical Sciences Research Council [EP/R513295/1].
CL is supported by the SNSF Grant [P2EZP2_188068].
HO is supported by the EPSRC grant “Datasig” [EP/S026347/1], the Alan Turing Institute, the Oxford-Man Institute, and the Centre for Intelligent Multidimensional Data Analysis (CIMDA).
Acknowledgments
HO would like to thank Manu Eder for helpful discussions. CL would like to thank Gudmund Pammer for pointing out the fact that the metric characterizes the convergence in when V is a locally compact space.
Citation
Patric Bonnier. Chong Liu. Harald Oberhauser. "Adapted topologies and higher rank signatures." Ann. Appl. Probab. 33 (3) 2136 - 2175, June 2023. https://doi.org/10.1214/22-AAP1862
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