Parameter Estimation for a Bidimensional Partially Observed Ornstein–Uhlenbeck Process with Biological Application

Abstract. We consider a bidimensional Ornstein–Uhlenbeck process to describe the tissue microvascularization in anti‐cancer therapy. Data are discrete, partial and noisy observations of this stochastic differential equation (SDE). Our aim is to estimate the SDE parameters. We use the main advantage of a one‐dimensional observation to obtain an easy way to compute the exact likelihood using the Kalman filter recursion, which allows to implement an easy numerical maximization of the likelihood. Furthermore, we establish the link between the observations and an ARMA process and we deduce the asymptotic properties of the maximum likelihood estimator. We show that this ARMA property can be generalized to a higher dimensional underlying Ornstein–Uhlenbeck diffusion. We compare this estimator with the one obtained by the well‐known expectation maximization algorithm on simulated data. Our estimation methods can be directly applied to other biological contexts such as drug pharmacokinetics or hormone secretions.