Intrinsic Location Parameter of a Diffusion Process
R. W. R. Darling, National Security Agency
Abstract
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.
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