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.

翻译：随机微分方程是物理学及科学领域中广泛使用的建模工具。在多数建模场景中，驱动动力学或运动的随机波动具有非平凡的时间相关结构，这使得随机微分方程成为非马尔可夫过程——这一现象通常被称为“有色”噪声。因此，发展能够从数学和数值角度有效研究（可能非马尔可夫的）随机微分方程的工具至关重要。本报告形式化了一套基于所谓“运动广义坐标”来分析和数值研究随机微分方程的数学理论。与粗糙路径理论类似，我们针对噪声的特定实现逐路径分析随机微分方程，而非仅基于概率分析。与成熟的马尔可夫实现理论一致，我们将非马尔可夫随机微分方程在扩展空间中实现为马尔可夫过程。然而，与经典马尔可夫实现理论不同，本理论中的马尔可夫实现在短时间尺度上精确，且在流和涨落为解析函数时可在全局时间上精确成立。该理论对于具有解析流和涨落的随机微分方程是精确的，当流和涨落可微时则提供近似。它提供了有用的分析工具，我们利用这些工具求解了具有解析涨落的线性随机微分方程。该理论也可能有助于研究更粗糙的随机微分方程，因为这些方程可视为光滑随机微分方程的极限。本理论为随机微分方程的模拟、滤波和控制提供了有效且计算简便的方法；其中，我们重新推导了广义贝叶斯滤波——一种时间序列分析的前沿方法。展望未来，本报告表明广义坐标在随机微分方程领域具有广泛的应用前景。