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<p>Diffusion processes are commonly used in the modeling of time series that exhibit stationarity and Markovianity. However, those two properties do not guarantee that a diffusive process is sufficient for the time series. In this paper, we develop a test for the sufficiency of a diffusion process for an observed time series. To develop the test we capitalize on the Kramers–Moyal (KM) expansion: a Taylor expansion of the integral form of the master equation that describes Markov continuous-time processes. In the idealized case, if the observed data indeed arise from a true diffusion process, then the KM expansion should truncate naturally after the second term. In theory, this means that any higher-order (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\varvec{\ge } \varvec{3}\)</EquationSource> </InlineEquation>) KM coefficients should be zero. However, in practice, the discrete nature of measurement introduces artificial higher-order KM coefficients, even when the underlying process is truly diffusive. Nonetheless, for genuinely diffusive systems, it is expected that the sampling distribution of a statistic associated with higher-order coefficients will be different than non-diffusive ones. This is a viable avenue for testing the appropriateness of a diffusion model given an observed time series. We take advantage of this and propose a meaningful statistic that could inform whether or not a diffusion model is sufficient for a given time series. We then build a test that involves reconstructing the diffusion equation to generate surrogate paths, yielding a bootstrap distribution against which the observed statistic could be compared. We evaluate the sensitivity and selectivity of the proposed test in Monte Carlo studies.</p>

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A Reconstruct-then-Bootstrap Test for the Sufficiency of Diffusion Processes

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  • 发表日期：

    2026年3月19日

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  • Kathleen Medriano,
  • Joachim Vandekerckhove

摘要

Diffusion processes are commonly used in the modeling of time series that exhibit stationarity and Markovianity. However, those two properties do not guarantee that a diffusive process is sufficient for the time series. In this paper, we develop a test for the sufficiency of a diffusion process for an observed time series. To develop the test we capitalize on the Kramers–Moyal (KM) expansion: a Taylor expansion of the integral form of the master equation that describes Markov continuous-time processes. In the idealized case, if the observed data indeed arise from a true diffusion process, then the KM expansion should truncate naturally after the second term. In theory, this means that any higher-order ( \(\varvec{\ge } \varvec{3}\) ) KM coefficients should be zero. However, in practice, the discrete nature of measurement introduces artificial higher-order KM coefficients, even when the underlying process is truly diffusive. Nonetheless, for genuinely diffusive systems, it is expected that the sampling distribution of a statistic associated with higher-order coefficients will be different than non-diffusive ones. This is a viable avenue for testing the appropriateness of a diffusion model given an observed time series. We take advantage of this and propose a meaningful statistic that could inform whether or not a diffusion model is sufficient for a given time series. We then build a test that involves reconstructing the diffusion equation to generate surrogate paths, yielding a bootstrap distribution against which the observed statistic could be compared. We evaluate the sensitivity and selectivity of the proposed test in Monte Carlo studies.