# Probability and Statistics Seminar

### Estimation of the drift of a $2n$-dimension OU process

A $2n$-dimension Ornstein-Uhlenbeck (OU) process for which the diffusion matrix is singular is considered. This process is used as a model for the dynamic behavior of vibrating engineering structures such as bridges, buildings, dams, among others. We study the problem of estimating the vibration frequencies of the structure or, equivalently, the parameters of the stochastic differential equation (SDE) that governs the OU process.

Firstly, it is considered the case where the OU process is perturbed by an independent wiener process. The maximum likelihood estimator of the drift matrix is obtained and the properties of the estimator are established. The local asymptotic normality of the estimator is analyzed in detail. Since general regularity conditions do not hold in this case (the diffusion matrix is singular), theoretical results from the classic literature on the subject do not immediately apply and an alternative approach based on the Laplace transform is used.

Secondly, it is considered the case where the OU process is perturbed by two independent fractional brownian motions. Models involving fractional noises have not been widely used in engineering. However, many problems in engineering involve processes exhibiting long memory. For this reason, the estimation of the parameters of multidimensional state space linear models, described by SDEs and disturbed by fractional Brownian motion, has a potential application in different areas of engineering. We analyze the problem of estimating the drift parameters of a $2$- dimension linear stochastic differential equation perturbed by two independent fractional Brownian motions with the same Hurst parameter belonging to $(1/2,1)$. The maximum likelihood estimator of the drift parameters is obtained after a transformation of the original model and making use of the so called fundamental martingale.

In both cases, a simulation study is presented in the context of a real world situation that illustrates the asymptotic behavior of the maximum likelihood estimator of the drift matrix.