We consider the problem of nonparametric estimation of the drift and diffusion coefficients of a Stochastic Differential Equation (SDE), based on $n$ independent replicates $\left\{X_i(t)\::\: t\in [0,1]\right\}_{1 \leq i \leq n}$, observed sparsely and irregularly on the unit interval, and subject to additive noise corruption. By sparse we intend to mean that the number of measurements per path can be arbitrary (as small as two), and remain constant with respect to $n$. We focus on time-inhomogeneous SDE of the form $dX_t = \mu(t)X_t^{\alpha}dt + \sigma(t)X_t^{\beta}dW_t$, where $\alpha \in \{0,1\}$ and $\beta \in \{0,1/2,1\}$, which includes prominent examples such as Brownian motion, Ornstein-Uhlenbeck process, geometric Brownian motion, and Brownian bridge. Our estimators are constructed by relating the local (drift/diffusion) parameters of the diffusion to their global parameters (mean/covariance, and their derivatives) by means of an apparently novel Partial Differential Equation (PDE). This allows us to use methods inspired by functional data analysis, and pool information across the sparsely measured paths. The methodology we develop is fully non-parametric and avoids any functional form specification on the time-dependency of either the drift function or the diffusion function. We establish almost sure uniform asymptotic convergence rates of the proposed estimators as the number of observed curves $n$ grows to infinity. Our rates are non-asymptotic in the number of measurements per path, explicitly reflecting how different sampling frequency might affect the speed of convergence. Our framework suggests possible further fruitful interactions between FDA and SDE methods in problems with replication.

翻译：我们研究基于$n$条独立重复、在单位区间上稀疏且不规则观测、并受加性噪声污染的路径$\left\{X_i(t)\::\: t\in [0,1]\right\}_{1 \leq i \leq n}$，对随机微分方程(SDE)漂移项与扩散项系数的非参数估计问题。所谓“稀疏”是指每条路径的测量次数可任意少（最低仅两次）且不随$n$增长。我们聚焦于形如$dX_t = \mu(t)X_t^{\alpha}dt + \sigma(t)X_t^{\beta}dW_t$的时间非齐次SDE，其中$\alpha \in \{0,1\}$且$\beta \in \{0,1/2,1\}$，涵盖布朗运动、Ornstein-Uhlenbeck过程、几何布朗运动与布朗桥等典型案例。通过引入一个全新的偏微分方程(PDE)，我们将扩散过程的局部（漂移/扩散）参数与其全局参数（均值/协方差及其导数）建立联系，由此构造估计量。该方法借助函数型数据分析思想，整合稀疏路径间的信息，完全无需对漂移函数或扩散函数的时间依赖性预设任何函数形式。我们证明了当观测曲线数$n$趋于无穷时，所提估计量达到几乎必然一致渐近收敛率。该收敛率在每条路径测量次数意义上具有非渐近特性，明确反映了不同采样频率对收敛速度的影响。本框架揭示了在含复制的SDE问题中，函数型数据分析与SDE方法可产生富有成效的交叉融合。