Le Cam–Stratonovich–Boole theory for Itô diffusions

Jaya P. N. Bishwal

doi:10.1515/rose-2023-2004

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Le Cam–Stratonovich–Boole theory for Itô diffusions

Jaya P. N. Bishwal

Veröffentlicht/Copyright: 28. Februar 2023

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Aus der Zeitschrift Random Operators and Stochastic Equations Band 31 Heft 2

Abstract

We connect the theory of local asymptotic normality (LAN) of Le Cam to Boole’s approximation of the Stratonovich stochastic integral by estimating the parameter in the nonlinear drift coefficient of an ergodic diffusion process satisfying a homogeneous Itô stochastic differential equation based on discretely spaced dense observations of the process. The asymptotic normality and local asymptotic minimaxity (in the Hajek–Le Cam sense) of approximate maximum likelihood estimators, approximate maximum probability estimators and approximate Bayes estimators based on Itô and Boole’s approximations of the continuous likelihood are obtained under an almost slowly increasing experimental design (ASIED) condition ( T n 6 / 7 → 0 as T → ∞ and n → ∞ , where T is the length of the observation time and n is the number of observations) through the weak convergence of the approximate likelihood ratio random fields. Among other things, the Bernstein–von Mises type theorems concerning the convergence of suitably normalized and centered approximate posterior distributions to normal distribution under the same design condition are proved. Asymptotic normality and asymptotic efficiency of the conditional least squares estimator under the same design condition are obtained as a by-product. The log-likelihood derivatives based on Itô approximations are martingales, but the log-likelihood derivatives based on Boole’s approximations are not martingales but weighted averages of forward and backward martingales. These new approximations have faster rate of convergence than the martingale approximations. The methods would have advantages over Euler and Milstein approximations for Monte Carlo simulations.

Keywords: Itô stochastic differential equation; Fisk–Stratonovich integral; Boole approximation; Monte Carlo method; diffusion process; discrete observations; forward-backward martingale approximation; almost slowly increasing experimental design; approximate maximum likelihood estimators; Bernstein–von Mises theorem; approximate posterior distributions; approximate Bayes estimators; approximate maximum probability estimators; conditional least squares estimator; weak convergence of random fields; local asymptotic normality; Hajek–Le Cam minimaxity

MSC 2010: 62F12; 62F15; 62M05; 60F05; 60F10; 60H10

Communicated by Nikolai Leonenko

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Received: 2021-12-25

Accepted: 2022-03-24

Published Online: 2023-02-28

Published in Print: 2023-06-01

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Artikel in diesem Heft

https://doi.org/10.1515/rose-2023-2004

Schlagwörter für diesen Artikel

Itô stochastic differential equation; Fisk–Stratonovich integral; Boole approximation; Monte Carlo method; diffusion process; discrete observations; forward-backward martingale approximation; almost slowly increasing experimental design; approximate maximum likelihood estimators; Bernstein–von Mises theorem; approximate posterior distributions; approximate Bayes estimators; approximate maximum probability estimators; conditional least squares estimator; weak convergence of random fields; local asymptotic normality; Hajek–Le Cam minimaxity