1998/09/06 by R. W. R. Darling, Darling, R. W. R.
Environmental Science · Mathematics · #58G32 #60H30 #Analysis of environmental and stochastic processes #Differential Equations and Numerical Methods #FOS: Mathematics #Optimization and Control (math.OC) #Probability (math.PR) #advanced mathematical theories #math.OC #math.PR #msc:58G32 #msc:60H30
paper · pdf · doi:10.48550/arxiv.math/9809027
25 pages, 1 figure
arxiv created 1998/09/06 · openalex publication_date 1998/09/06 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For nonlinear functions f of a random vector Y, E[f(Y)] and f(E[Y]) usually differ. Consequently the mathematical expectation of Y is not intrinsic: when we change coordinate systems, it is not invariant.This article is about a fundamental and hitherto neglected property of random vectors of the form Y = f(X(t)), where X(t) is the value at time t of a diffusion process X: namely that there exists a measure of location, called the "intrinsic location parameter" (ILP), which coincides with mathematical expectation only in special cases, and which is invariant under change of coordinate systems. The construction uses martingales with respect to the intrinsic geometry of diffusion processes, and the heat flow of harmonic mappings. We compute formulas which could be useful to statisticians, engineers, and others who use diffusion process models; these have immediate application, discussed in a separate article, to the construction of an intrinsic nonlinear analog to the Kalman Filter. We present here a numerical simulation of a nonlinear SDE, showing how well the ILP formula tracks the mean of the SDE for a Euclidean geometry.