分数交互粒子系统的非参数估计

Nonparametric Estimation for FractionalInteracting Particle Systems

摘要: 在本文中，我们考虑以下交互粒子系统的非参数估计问题: dX_t^θ,i,N=∑_m=1^pθ_i,mb_m(X_t^θ,i,N,μt^θ,N)dt+σdBt^H,i,i=1,...,N，t∈[0,T],

Abstract: 其初始分布为L(X₀^θ,1,N,...,X₀^θ,N,N)。其中θ_i=(θ_i,1,θ_i,p)为未知参数向量，过程(B_t^H,i)t∈[0,T],i=1,...,N是独立的实值 Hurst系数为H ∈(1/2,1)的分数布朗运动，且与系统的初始值(X₀^θ,1,N,...,X₀^θ,N,N)相互独立，σ > 0为波动率参数，μ_t^θ,N为系统在时刻t的经验测度。我们采用矩估计方法分别得到∑_m=1^pθ_i,mb_m(X_t^θ,i,N, μt^θ,N)(定理1.1)、θ_i(i = 1,...,N)(定理1.2)以及θ_i(i= 1,.....N)的公共密度函数f(定理1.3）的估计量。研究结果表明，收敛速率同时依赖于分数布朗运动的 Hurst系数H与噪声参数σ。In this paper, we consider the nonparametric estimation for the following interacting particle system, dX_t^θ,i,N=∑_m=1^pθ_i,mb_m(X_t^θ,i,N,μt^θ,N)dt+σdBt^H,i,i=1,...,N，t∈[0,T], with initial distribution L(X₀^θ,1,N,...,X₀^θ,N,N),where θ_i=(θ_i,1,θ_i,p) is an unknown parameter vector, the processes (B_t^H,i)t∈[0,T],i=1,...,N, are independent ℝ-valued fractional Brownian motions of Hurst index H ∈(1/2,1),independent of the initial value (X₀^θ,1,N,...,X₀^θ,N,N)of the system, σ > 0 is a volatility parameter, and μ_t^θ,N is the empirical measure of the system at time t. We use the moment method to obtain the estimators for ∑_m=1^pθ_i,mb_m(X_t^θ,i,N, μt^θ,N)(Theorem1.1)、θ_i(i = 1,...,N)(Theorem 1.2)and the common density function f of θ_i(i= 1,.....N)(Theorem 1.3).Our results demonstrate that the convergence rates depend on both the Hurst index H of the fBm and the noise parameter σ.