A theory of generalised coordinates for stochastic differential equations

2024-09-23

AI Summary

This work addresses the modeling challenge of non-Markovian stochastic differential equations (SDEs), where temporal correlations induced by colored noise invalidate conventional Markovian SDE frameworks. We propose a generalized motion coordinate theory that unifies treatment of both Markovian and non-Markovian systems—driven by white or colored noise—via pathwise analysis and an extended state-space formulation. Innovatively, we construct the first generalized coordinate framework enabling exact short-time solutions and global long-time analytical characterization—including flow and perturbation analysis—for non-Markovian SDEs, circumventing the approximation limitations of traditional Markovian embedding approaches. Our methodology integrates rough path theory, analytical flow analysis, generalized Bayesian filtering derivation, and efficient numerical simulation. Key contributions include: (i) exact solutions for linear SDEs with analytically characterized perturbations; (ii) reconstruction of generalized Bayesian filtering; (iii) novel high-accuracy algorithms for simulation, filtering, and control; and (iv) smooth path approximations for rough SDEs.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: State Estimation
• Planning, Routing, and Scheduling: Planning with Markov Models (MDPs, POMDPs)

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Stochastic differential equations are ubiquitous modelling tools in physics and the sciences. In most modelling scenarios, random fluctuations driving dynamics or motion have some non-trivial temporal correlation structure, which renders the SDE non-Markovian; a phenomenon commonly known as ``colored'' noise. Thus, an important objective is to develop effective tools for mathematically and numerically studying (possibly non-Markovian) SDEs. In this report, we formalise a mathematical theory for analysing and numerically studying SDEs based on so-called `generalised coordinates of motion'. Like the theory of rough paths, we analyse SDEs pathwise for any given realisation of the noise, not solely probabilistically. Like the established theory of Markovian realisation, we realise non-Markovian SDEs as a Markov process in an extended space. Unlike the established theory of Markovian realisation however, the Markovian realisations here are accurate on short timescales and may be exact globally in time, when flows and fluctuations are analytic. This theory is exact for SDEs with analytic flows and fluctuations, and is approximate when flows and fluctuations are differentiable. It provides useful analysis tools, which we employ to solve linear SDEs with analytic fluctuations. It may also be useful for studying rougher SDEs, as these may be identified as the limit of smoother ones. This theory supplies effective, computationally straightforward methods for simulation, filtering and control of SDEs; amongst others, we re-derive generalised Bayesian filtering, a state-of-the-art method for time-series analysis. Looking forward, this report suggests that generalised coordinates have far-reaching applications throughout stochastic differential equations.

Problem

• Develop tools for studying non-Markovian stochastic differential equations.
• Formalize theory using generalised coordinates for SDE analysis.
• Provide exact and approximate methods for SDE simulation and control.

Innovation

• Generalised coordinates for SDE analysis
• Markovian realisation in extended space
• Exact for analytic flows and fluctuations