Arcade Processes for Informed Martingale Interpolation and Transport
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 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 nearly-Markov randomized arcade processes, the dynamics of FAMs are informed through Bayesian updating. FAMs can be connected to martingale optimal transport (MOT) by considering optimally-coupled target random variables. Moreover, FAMs allow to formulate the information-based martingale optimal transport problem, which enables the introduction of noise in MOT, in a similar fashion to how Schrödinger's problem introduces noise in optimal transport. This information-based transport problem 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.