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 \textit{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 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:我们考虑的是,根据美元独立复制 $\ left\\ x_ x_i( t)\ :\: t\ in [0,1\\right\1\\leq i\leq n}$,在单位间隔上很少和不定期地观察到,并受到添加噪音腐败的影响。我们打算通过\ textit{sparse} 来表示,每个路径的测量数量可以是任意的(小于2),并且对于美元保持恒定。我们侧重于以美元独立复制的 $left\\ xx_x_i_i( t)\leq\ i\leq\\\ i\ i\ leq n}:\ : t\ 在单位间隔中, 很少或不定期地观察到, 以美元表示 0. 1 美元和 $\beta 格式的测量数量, 包括不以布朗运动、 Ornstein- Ulelex 和 blicker 的计算方法 。