Adaptive inference for small diffusion processes based on sampled data

Abstract

We consider parametric estimation for multi-dimensional diffusion processes with a small dispersion parameter ε from discrete observations. For parametric estimation of diffusion processes, the main targets are the drift parameter α and the diffusion parameter β. In this paper, we propose two types of adaptive estimators for (α, β) and show their asymptotic properties under ε → 0, n → ∞ and the balance condition that (εn ρ ) −1 = O(1) for some ρ ≥ 1/2. In simulation studies, we examine and compare asymptotic behaviors of the two kinds of adaptive estimators. Moreover, we treat the SIR model which describes a simple epidemic spread for a biological application.

Parameter estimation for a linear parabolic SPDE model in two space dimensions with a small noise

“…, n, where ∆ n = 1/n. Parameter estimation for diffusion processes with a small noise based on discrete observations has been studied by many researchers, see Genon-Catalot [8], Laredo [21], Sørensen and Uchida [29], Uchida [31], [32], Gloter and Sørensen [10], Guy et al [11], Nomura and Uchida [26], Kaino and Uchida [17] Kawai and Uchida [20] and reference therein. In particular, Uchida [31] estimated a parameter appearing in both the drift and diffusion coefficients under (n ) −1 = O(1), and Gloter and Sørensen [10] considered joint estimation for both the drift and diffusion parameters under (n ρ ) −1 = O(1) for some ρ > 0.…”

“…Since the coordinate process of the SPDE (1.1) is a diffusion process, we derive an estimator of θ 0 based on statistical inference for diffusion processes with a small noise in an analogous manner to Kaino and Uchida [19]. For statistical inference for diffusion processes with a small noise based on discrete observations, see Genon-Catalot [8], Laredo [21], Sørensen and Uchida [29], Uchida [31], [32], Gloter and Sørensen [10], Guy et al [11], Nomura and Uchida [26], Kaino and Uchida [17] and Kawai and Uchida [20].…”