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Intrinsic Location Parameter of a Diffusion Process


 
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1. Title Title of document Intrinsic Location Parameter of a Diffusion Process
 
2. Creator Author's name, affiliation, country R. W. R. Darling; National Security Agency
 
3. Subject Discipline(s) Mathematics
 
3. Subject Keyword(s) intrinsic location parameter, gamma-martingale, stochastic differential equation, forward-backwards SDE, harmonic map, nonlinear heat equation
 
3. Subject Subject classification 60H30, 58G32.
 
4. Description 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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7. Date (YYYY-MM-DD) 2001-03-14
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/102
 
10. Identifier Digital Object Identifier 10.1214/EJP.v7-102
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 7
 
12. Language English=en en
 
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