Estimation of the parameters of the Ornstein-Uhlenbeck's stochastic process

Abstract

It is considered Ornstein-Uhlenbeck process $ x_t = x_0 e^{-θt} + μ(1-e^{-θt}) + σ\int_0^t e^{-θ(t-s)} dW_s$, where $x_0 \in R$, $θ>0$, $ μ\in R$ and $σ> 0$ are parameters. By use values $(z_k)_{k \in N}$ of corresponding trajectories at a fixed positive moment $t$, a consistent estimate of each unknown parameter of the Ornstein-Uhlenbeck's stochastic process is constructed under assumption that all another parameters are known.