2022/10/07 by David Sabin-Miller, Daniel M. Abrams, Sabin-Miller, David +1 · 1 citation
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Adaptation and Self-Organizing Systems (nlin.AO) #Advanced Thermodynamics and Statistical Mechanics #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Biology Tumor Growth #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2210.03781
openalex publication_date 2022/10/07 · openalex created_date 2022/10/12 · openalex updated_date 2026/07/28
Many real-world systems exhibit ``noisy'' evolution in time; interpreting their finitely-sampled behavior as arising from continuous-time processes (in the Itô or Stratonovich sense) has led to significant success in modeling and analysis in a wide variety of fields. Yet such interpretation hinges on a fundamental linear separation of randomness from determinism in the underlying dynamics. Here we propose some theoretical systems which resist easy and self-consistent interpretation into this well-defined class of equations, requiring an expansion of the interpretive framework. We argue that a wider class of stochastic differential equations, where evolution depends nonlinearly on a random or effectively-random quantity, may be consistently interpreted and in fact exhibit finite-time stochastic behavior in line with an equivalent Itô process, at which point many existing numerical and analytical techniques may be used. We put forward a method for this conversion, and demonstrate its use on both a toy system and on a system of direct physical relevance: the velocity of a meso-scale particle suspended in a turbulent fluid. This work enables the theoretical and numerical examination of a wide class of mathematical models which might otherwise be oversimplified due to a lack of appropriate tools.