Nonlinear Analysis 63 (2005) e2367 – e2375 www.elsevier.com/locate/na Bayesian parameter estimation and prediction in mean reverting stochastic diffusion models  Bevan Thompson ∗ , Igor Vladimirov Department of Mathematics, SPS, The University of Queensland, Brisbane, QLD 4072, Australia Abstract We consider the problem of Bayesian parameter estimation and prediction in a diffusion process governed by an Ito stochastic differential equation. The diffusion coefficient function is assumed known while the drift term is an affine function of the state with unknown slope and free coefficients which are assigned a bivariate Gaussian prior distribution. We derive closed-form expressions for the Bayesian predictor of the underlying process and for its mean square accuracy and discuss the conditions under which the posterior parameter uncertainty can be neglected.  2005 Elsevier Ltd. All rights reserved. MSC: 60G25; 60H10; 62F15 Keywords: Stochastic differential equation; Bayesian prediction; Conjugate prior 1. Introduction The idea of predicting a random process on the basis of model identification, including parameter estimation, in the framework of the Bayesian theory [4] is widely known in adaptive control [8,11]. Having numerous electrical engineering applications, the Bayesian approach in econometrics coexists with maximum likelihood (ML)-based methods. The study of the ML estimators of the drift parameters of stochastic differential equations (SDEs)  The work is supported by the Australian Research Council SPIRT Grant C00106980 and Tarong Energy Corporation. ∗ Corresponding author. Fax: +61 7 33651477. E-mail addresses: hbt@maths.uq.edu.au (B. Thompson), igv@maths.uq.edu.au (I. Vladimirov). 0362-546X/$ - see front matter  2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.02.095