APPROXIMATE MAXIMUM LIKELIHOOD METHOD FOR ESTIMATING A TWO-THRESHOLD ORNSTEIN-ULENBECK PROCESS

Abstract

A two-regime two-threshold process allows modeling complex systems in which dynamics change upon reaching threshold levels. This paper proposes an approximate maximum likelihood method for estimating the parameters of a two-threshold regime diffusion process with discrete sample data based on approximating the logarithmic likelihood function of observations. The logarithmic form of the likelihood function improves computational stability, which is particularly important for threshold models with a large number of parameters. The discrete model is built based on the Ornstein- Uhlenbeck process, defined by the corresponding stochastic differential equation and its subsequent discretization using the Euler scheme, which is simple to implement and ensures the required accuracy with an appropriately chosen time step. The Ornstein-Uhlenbeck process is convenient and widely used in various applications because it is Gaussian, and its stationarity condition is easily formulated, which allows for working with data in the form of a time series. By differentiating the likelihood function with respect to each parameter, we obtain a series of equations for determining the estimates of the shift, diffusion, and threshold parameters. The studied two-threshold regime process behaves differently at values below the first threshold, between the thresholds, and above the second threshold. In each of these intervals, the process may exhibit different parameter behavior. In practical applications, it is crucial to obtain the most accurate estimates for the drift, diffusion, and threshold parameters, as the precision of these estimates affects the model’s ability to accurately describe the process dynamics. The paper also proposes a computational algorithm for the two-threshold regime model. The model divides the observations into several ranges according to the thresholds r1 and r2. Each range is described by its parameters, allowing for the consideration of different process behaviors within each interval. Within each range, the likelihood function is calculated, reflecting the probability of obtaining the observed data, given the correctness of the parameters in each range. This approach provides the model with flexibility for analyzing complex stochastic processes with threshold effects, particularly in financial markets, where asset price changes can significantly depend on reaching specific thresholds, aligning with market strategies.