Large deviations for squares of Bessel and Ornstein-Uhlenbeck processes
This paper is concerned with the Large Deviations (LD) behaviour of squared Bessel processes (as well as a few related processes) considered over a finite time-horizon, when the dimensional parameter of the Bessel process tends to infinity. \newline First of all, fixing a space \(C^{\alpha}[0,T]\) of \(\alpha\)-Hölder continuous functions (where \(0<\alpha<1/2\)), the exponential tightness of the corresponding probability distributions is established. Several proofs are then given for the LD upper and lower bounds: a first method of proof consists in using the properties of an auxiliary exponential martingale, a second method of proof relies on an additivity property for squared Bessel processes and uses Cramér's theorem; the advantage of this second approach lies in the fact that it should be applicable also for other families of Markov processes having a similar additivity property, e.g. continuous state branching processes. The authors also show that the Freidlin-Wentzell approach may be extended to the present context, even though the diffusion coefficient in the stochastic differential equation of squared Bessel processes fails to be Lipschitz continuous. These different approaches lead to several variational expressions for the rate function of the LD principle, and these variational expressions are compared in a separate section. LD results are also derived for other related processes (squared radial Ornstein-Uhlenbeck processes, squared Bessel bridges), using essentially the contraction principle.
scientific article; zbMATH DE number 2106563