Abstract
In this work, an attempt is made to estimate time varying parameters in a linear stochastic differential equation. By defining $m_{k}$ as the local admissible sample/data observation size at time $t_{k}$, parameters and state at time $t_{k}$ are estimated using past data on interval $[t_{k-m_{k}+1}, t_{k}]$. We show that the parameter estimates at each time $t_{k}$ converge in probability to the true value of the parameters being estimated. A numerical simulation is presented by applying the local lagged adapted generalized method of moments (LLGMM) method to the stochastic differential models governing prices of energy commodities and stock price processes.
Highlights
In this work, we estimate the time varying parameters in a linear stochastic differential equation (SDE) in a systemic and unified way
We studied two special linear stochastic differential equations, namely the geometric brownian motion and the Ornstein-Uhlenbeck time varying stochastic differential equation
We note here that the first case is the generalization of the geometric stochastic differential equation with time varying parameters
Summary
We estimate the time varying parameters in a linear stochastic differential equation (SDE) in a systemic and unified way. Utilizing Monte-Carlo method and the Euler-type (Kloeden & Platen, 1992) stochastic discretization scheme, we construct systems of local moments/observation equations based on the number of parameters present. Using the method of moments (Casella & Berger, 2002) in the context of lagged adaptive expectation process (Paothong & Ladde, 2013), we describe theoretical parametric estimation procedure for the SDE.
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More From: International Journal of Statistics and Probability
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