Replication and Its Application to Weak Convergence

TL;DR

This paper introduces a replication methodology for stochastic processes on compact metric spaces, simplifying proofs of convergence, tightness, and compactness, and enabling broader applications in stochastic analysis.

Contribution

The paper develops a novel replication technique that transfers properties of stochastic objects to compact metric spaces, facilitating proofs and extending classical results to general topological spaces.

Findings

Finite-dimensional convergence for processes on topological spaces

New tightness and compactness criteria for Skorokhod space

Applications to martingale problems and filtering equations

Abstract

Herein, a methodology is developed to replicate functions, measures and stochastic processes onto a compact metric space. Many results are easily established for the replica objects and then transferred back to the original ones. Two problems are solved within to demonstrate the method: (1) Finite-dimensional convergence for processes living on general topological spaces. (2) New tightness and relative compactness criteria are given for the Skorokhod space $D(\mathbf{R}^{+};E)$ with $E$ being a general Tychonoff space. The methods herein are also used in companion papers to establish the: (3) existence of, uniqueness of and convergence to martingale problem solutions, (4) classical Fujisaki-Kallianpur-Kunita and Duncan-Mortensen-Zakai filtering equations and stationary filters, (5) finite-dimensional convergence to stationary signal-filter pairs, (6) invariant measures of Markov processes, and (7) Ray-Knight theory all in general settings.