A probabilistic approach to drift estimation from stochastic data

TL;DR

An interpretation of a drift as a conditional expectation is described, which makes its estimation feasible via kernel-integral methods and some techniques are proposed to overcome the challenge of dimensionality if the stochastic differential equations carry some structure enabling sparsity.

Abstract

Timeseries generated from a dynamical source can often be modeled as sample paths of a stochastic differential equation (SDE). The timeseries thus reflects the motion of a particle which flows along the direction provided by a drift / vector field, and is simultaneously scattered by the effect of white noise. The resulting motion can only be described as a random process instead of a solution curve. Due to the non-deterministic nature of this motion, the task of determining the drift from data is quite challenging, since the data does not directly represent the directional information of the flow. This paper describes an interpretation of a drift as a conditional expectation, which makes its estimation feasible via kernel-integral methods. In addition, some techniques are proposed to overcome the challenge of dimensionality if the SDE's carry some structure enabling sparsity. The technique is shown to be convergent, consistent and permits a wide choice of kernels.