A theory of generalised coordinates for stochastic differential equations

TL;DR

This work develops a theory of generalised coordinates for stochastic differential equations, enabling pathwise, Taylor-expansion based analysis and Markovian realisations in an extended state space. The framework is exact for SDEs with analytic flows and noise, and provides accurate short-time approximations otherwise, linking to Wong–Zakai results and rough-path ideas. It yields practical methods for simulation, density dynamics, and Bayesian filtering, including a generalised filtering scheme that uses a Laplace approximation in generalized coordinates. By embedding higher-order derivatives as coordinates, the approach offers a unifying view of analytic approximation, path integrals, and numerical schemes, with demonstrated utility in filtering and most-likely-path computations. The work points to broad applications across stochastic modelling, control, and time-series analysis where colored noise and non-Markovian dynamics are essential.

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.

Figures (5)

• Figure 1: Numerical integration of 1 and 2 dimensional linear SDEs. On the left (resp. right) panels, we simulate a one (resp. two) dimensional linear SDE driven by a stationary Gaussian process with Gaussian autocovariance (white noise convolved with a Gaussian). For the top panels, we sampled white noise paths, convolved them with a Gaussian, and numerically integrated the resulting SDE pathwise using Euler's method, providing a 'ground truth' comparative baseline. For the bottom panels, we used the zigzag numerical integration method (both zigzag methods coincide for this system), plotting an exact Taylor expansion of solution sample paths that uses derivatives up to order $10$. On the left, each (1d) sample path is represented as a function of time in a different colour, while on the right each (2d) sample path is plotted on a plane with a start in magenta and an end in cyan. The path of least action (i.e. in the absence of noise) is plotted in each panel in a dotted grey line. Note that the trajectories in the top and bottom panels cannot be compared individually---only statistically---since they correspond to distinct noise realisations (since the noise samples are generated differently in each method).
• Figure 2: Numerical integration of stochastic Lotka-Volterra system. We simulate a stochastic Lotka-Volterra system driven by a stationary Gaussian process with Gaussian autocovariance (white noise convolved with a Gaussian). The Lotka-Volterra system is a toy model for the relative proportions of predator and prey in an ecosystem Mao_2002. In the top panels we plot sample trajectories of the system over time, with the relative proportions of predators in red and prey in blue; in the bottom panels we plot these trajectories on a plane with a start in magenta and an end in cyan. The path of least action (i.e. in the absence of noise) is plotted in each panel as a dotted line. In the first column, we plot trajectories generated with Euler's method, providing a 'ground truth' comparative baseline. In the second and third columns, we plot trajectories generated with the zigzag and linearised zigzag methods. The zigzag and linearised zigzag methods solved the system for the same set of noise sample paths, while Euler's method used a different set of noise samples (since the noise is realised differently in this method).
• Figure 3: Numerical integration of stochastic Lorenz system. We simulate a stochastic Lorenz system driven by a stationary Gaussian process with Gaussian autocovariance (white noise convolved with a Gaussian). The plots are sample trajectories at various initialisations (shared across panels), in 3d in the top panels, with their 2d projections on the $x$-$y$ plane in the bottom panels. The first column shows paths of least action (i.e. in the absence of noise). The subsequent columns shows paths integrated with Euler, zigzag and linearised zigzag methods, respectively, where Euler serves as a comparative baseline. Importantly, the noise realisations were shared only by the zigzag and linearised zigzag methods (since the noise is realised differently in Euler's method).
• Figure 4: Recovering path of least action for linear SDE.Left: We show the true path of least action of a given 2 dimensional linear SDE with initial condition $(10,10)$ in a dashed grey line. We then show the solution to \ref{['eq: regularised equation for least action']} for $N+1=4$ orders of motion and different weightings $\lambda=1,10,100$, confirming the claim that it tends to the path of least action as $\lambda$ grows large. Right: We plot the Lagrangian over time ($+1$ to plot in log-scale) for solutions to \ref{['eq: regularised equation for least action']} corresponding to respective weightings of $\lambda$. The closer the solution is to the path of least action the more the Lagrangian is minimised at all times, and vice-versa.
• Figure 5: Generalised filtering for the Lorenz system. This Figure illustrates a simulation of generalised filtering on a partially observed stochastic Lorenz system, in a regime of low and high observation noise (left and right panel respectively). In the top panels, we see the one-dimensional time-series of noiseless observations in dashed black, corresponding to the sum of the three latent states. The data that were fed to the generalised filter comprised the measured time-series in dashed purple---this is the dashed black time-series plus observation noise. In the middle panels, there are three dashed black time-series corresponding to the latent states, which evolve according to a stochastic Lorenz system. In colour (yellow, red, blue) are the filtered time-series; that is, the most likely latent states at each time according to the approximate posterior belief $q$. In the top panel, the cyan time-series are filtered observation time-series, obtained as the sum of the coloured time-series in the middle panels. The bottom panel scores the Euclidean distance between filtered time-series in latent space and true time-series.

Theorems & Definitions (47)

• Definition 1.1.2: Generalised flow
• Remark 1.2.1: Faithfulness under the local linear approximation
• Theorem 1.2.2
• proof
• Definition 1.2.3
• Theorem 1.2.4: Cauchy–Kovalevskaya
• proof
• Definition 1.3.1: White noise, informal
• Lemma 1.3.2
• proof