Arcade Processes for Informed Martingale Interpolation

TL;DR

This paper introduces arcade processes and their randomised variants as a novel method for strong stochastic interpolation, constructing martingales that match target distributions with potential applications in optimal transport.

Contribution

It develops filtered arcade martingales (FAMs) as a new class of solutions for martingale interpolation problems, linking stochastic filtering with process construction.

Findings

Explicit examples and simulations of RAPs and FAMs are provided.

FAMs are almost-sure solutions to the martingale interpolation problem.

Potential connections to martingale optimal transport are discussed.

Abstract

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 randomisation allows one to match any finite sequence of target random variables, indexed by the given fixed dates, on the whole probability space. The randomised arcade processes (RAPs) can thus be interpreted as a generalisation of anticipative stochastic bridges. The filtrations generated by these processes are utilised to construct a class of martingales that 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 randomised 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. Several explicit examples for RAPs and FAMs are provided and simulated. This paper concludes with an outlook on potential connections between FAMs and martingale optimal transport, and related applications.