Arcade Processes for Informed Martingale Interpolation

Arcade processes are a class of continuous stochastic processes that interpolate in a strong sense, i.e., omega by omega, between zeros at fixed pre-specified times. Their additive randomization allows one to match any finite sequence of target random variables, indexed by the given fixed dates, on the whole probability space. The randomized arcade processes (RAPs) can thus be interpreted as a generalization of anticipative stochastic bridges. The filtrations generated by these processes are utilized to construct a class of martingales which interpolate between the given target random variables. These so-called filtered arcade martingales (FAMs) are almost-sure solutions to the martingale interpolation problem and reveal an underlying stochastic filtering structure. In the special case of conditionally-Markov randomized arcade processes, the dynamics of FAMs are informed by Bayesian updating. The same ideas are applied to filtered arcade reverse-martingales, which are constructed in a similar fashion, using reverse-filtrations of RAPs, instead. As a potential application of this theory, optimal transportation is explored: FAMs may be used to introduce noise in martingale optimal transport, in a similar fashion to how Schr\"odinger's problem introduces noise in optimal transport. This information-based approach to transport is concerned with selecting an optimal martingale coupling for the target random variables under the influence of the noise that is generated by an arcade process, and suggests application in finance or climate science, for instance.