An improved two-step method in stochastic differential equation's structural parameter estimation

Abstract

Non-parametric modelling is a method which relies heavily on data and motivated by the smoothness properties in estimating a function which involves spline and non-spline approaches. Spline approach consists of regression spline and smoothing spline. Regression spline characterised by the truncated power series basis with Bayesian approach is considered in the first step of a two-step method for estimating the structural parameters for stochastic differential equation (SDE). Previous methodology revealed the selection of knot and order of spline can be done heuristically based on a scatter plot. To overcome the subjective and tedious process of selecting the optimal knot and order of spline, an algorithm is proposed. A single optimal knot is selected out of all the points with exception of the first and the last data and the least value of Generalised Cross Validation is calculated for each order of spline. The spline model is later utilised in the second step to estimate the stochastic model parameters. In the second step, a non-parametric criterion is proposed for estimating the diffusion parameter of SDE. Linear and non-linear SDE consisting of Geometric Brownian Motion (GBM) for the former and logistic together with Lotka Volterra (LV) model for the later are tested using the two-step method for both simulated and real data. The results show high percentage of accuracy with 99.90% and 96.12% are obtained for GBM and LV model respectively for diffusion parameters of simulated data. This verifies the viability of the two-step method in the estimation of diffusion parameters of SDE with an improvement of a single knot selection.