[Paper Review] Nonparametric estimation for fractional diffusion processes with random effects

Mohamed El Omari, Hamid El Maroufy+1 more|arXiv (Cornell University)|Jan 16, 2019

Stochastic processes and financial applicationsEconomics, Econometrics and Finance23 references3 citations

TL;DR

This paper proposes nonparametric kernel and histogram estimators for the density of random effects in fractional stochastic differential equations (FSDEs) with long-range dependence, using observations from $N$ subjects over time horizon $T$. Under $H > 1/2$, the authors establish $L^2$-risk for kernel estimators and $L^1$-risk for histogram estimators as both $N$ and $T$ tend to infinity, providing the first nonparametric inference framework for FSDEs with random effects.

ABSTRACT

We propose a nonparametric estimation for a class of fractional stochastic differential equations (FSDE) with random effects. We precisely consider general linear fractional stochastic differential equations with drift depending on random effects and non-random diffusion. We build ordinary kernel estimators and histogram estimators and study their Lp-risk (p =1 or 2), when H>1/2. Asymptotic results are evaluated as both T = T(N) and N tend to infinity.

Motivation & Objective

• To address the lack of nonparametric inference methods for random effects in fractional stochastic differential equations (FSDEs) with long-range dependence.
• To develop consistent density estimators for the common density $f$ of i.i.d. random effects $\phi_i$ observed through $N$ sample paths of FSDEs.
• To analyze the asymptotic $L^p$-risk ($p=1,2$) of kernel and histogram estimators under joint asymptotics $N,T \to \infty$.
• To extend existing parametric inference for SDEs with random effects to the nonparametric setting driven by fractional Brownian motion.
• To provide theoretical guarantees for density estimation in models with unknown drift $a(\cdot)$ and known $b(\cdot), \sigma(\cdot)$.

Proposed method

• Propose ordinary kernel estimators for the density $f$ of random effects $\phi_i$, based on the empirical distribution of estimated effects $\widetilde{\phi}_{i,T}$ derived from $N$ sample paths of the FSDE.
• Derive $L^2$-risk bounds for kernel estimators under the assumption that $a(\cdot)$ is known or unknown, with $H \in (1/2,1)$.
• Construct histogram estimators for $f$ under the assumption that $f$ has compact support, simplifying technical analysis and reflecting practical relevance.
• Analyze the $L^1$-risk of histogram estimators under joint asymptotics $N,T \to \infty$, with $T = T(N)$.
• Use stochastic calculus for fractional Brownian motion and concentration inequalities to control the estimation error in the presence of long-range dependence.
• Establish consistency and convergence rates for both estimators by bounding the bias and variance components under regularity conditions on $b(t), \sigma(t)$, and $a(\cdot)$.

Experimental results

Research questions

• RQ1 Can nonparametric density estimators be consistently constructed for random effects in FSDEs with $H > 1/2$?
• RQ2 What are the $L^2$-risk properties of kernel estimators for the density $f$ of random effects when both $N$ and $T$ grow?
• RQ3 How do histogram estimators perform in terms of $L^1$-risk for compactly supported densities in the same asymptotic regime?
• RQ4 How does the presence of an unknown drift $a(\cdot)$ affect the estimation risk of the random effects density?
• RQ5 What are the theoretical limitations of extending the method to $H < 1/2$?

Key findings

• The $L^2$-risk of the kernel estimator converges to zero as both $N$ and $T$ tend to infinity, under regularity conditions on $b(t), \sigma(t)$, and $a(\cdot)$.
• For the kernel estimator, the convergence rate depends on the smoothness of the density $f$, the Hurst index $H$, and the choice of bandwidth $h$.
• The $L^1$-risk of the histogram estimator is controlled under the assumption that the density $f$ has compact support, enabling simpler technical analysis.
• The proposed estimators are consistent under joint asymptotics $N,T \to \infty$, even when the drift $a(\cdot)$ is unknown.
• The method breaks down for $H < 1/2$ due to the failure of a key moment condition required for consistency of the estimated random effects $\widetilde{\phi}_{i,T}$.
• Numerical simulations confirm the theoretical convergence of kernel and histogram estimators toward the true density across different $H$ values and random effect distributions (Gaussian and Gamma).