Nonparametric estimation for stochastic differential equations

Abstract

The aim of our work is to estimate the drift and diffusion coefficients of stochastic differential equations (SDE) nonparametrically using n i.i.d replicates, {Xi(t) : t ∈ [0, 1]}1≤i≤n, which are prone to additive noise corruption and are observed sparsely and erratically on the interval [0,1]. The word ”sparse” suggests that the number of measurements per path is arbitrary, possibly as low as two, and that they stay constant with respect to n. For the estimation problem, imposing the assumption of smoothness to use smoothing techniques that further annihilate noise leads to the exclusion of a range of stochastic processes including the diffusion process. However, the estimators used in this thesis allow the functional data analysis of the processes that have nowhere differentiable sample paths, even if the observations are discrete and include noise. We talk about, dX(t) = μ(t)(X(t))αdt + σ(t)(X(t))βdB(t) where α ∈ {0, 1} and β ∈ {0, 1/2, 1}. The time-inhomogeneous SDE is one way to represent this. Using systems of PDEs, the estimators have been built by connecting the local diffusion parameters to the global parameters. This approach is entirely nonparametric and is motivated by functional data analysis.