Free Ornstein-Uhlenbeck processes

Free Ornstein-Uhlenbeck processes are studied in finite von Neumann algebras. It is shown that a free self-decomposable probability measure on R can be realized as the distribution of a stationary free Ornstein-Uhlenbeck process driven by a free Levy process. A characterization of a probability measure on R to be the stationary distribution of a periodic free Ornstein-Uhlenbeck process driven by a free Levy process is given in terms of the Levy measure of the measure. Finally, the notion of a free fractional Brownian motion is introduced. It is proved that the free stochastic differential equation driven by a fractional free Brownian motion has a unique solution. We call the solution a fractional free Ornstein-Uhlenbeck process.     

Large deviations for squares of Bessel and Ornstein-Uhlenbeck processes

Let (X_t⁽⁾,t0) be the BESQ process starting at x. We are interested in large deviations as for the family {^–1X_t⁽⁾,tT}, – or, more generally, for the family of squared radial OU process. The main properties of this family allow us to develop three different approaches: an exponential martingale method, a Cramér–type theorem, thanks to a remarkable additivity property, and a Wentzell–Freidlin method, with the help of McKean results on the controlled equation. We also derive large deviations for Bessel bridges.Mathematics Subject Classification (2000): 60F10, 60J60     

Least squares estimator of Ornstein-Uhlenbeck processes driven by fractional Lévy processes with periodic mean

We deal with the least squares estimator for the drift parameters of an Ornstein-Uhlenbeck process with periodic mean function driven by fractional Lévy process. For this estimator, we obtain consistency and the asymptotic distribution. Compared with fractional Ornstein-Uhlenbeck and Ornstein-Uhlenbeck driven by Lévy process, they can be regarded both as a Lévy generalization of fractional Brownian motion and a fractional generaliza- tion of Lévy process.     

Ratio limit theorems for branching Ornstein-Uhlenbeck processes

We prove ratio limit theorems for critical ano supercritical branching Ornstein-Uhlenbeck processes. A finite first moment of the offspring distribution {p_n} assures convergence in probability for supercritical processes and conditional convergence in probability for critical processes. If even Σp_nnlog⁺log⁺n< ∞, then almost sure convergence obtains in the supercritical case.     

Maximum likelihood estimation for noncausal autoregressive processes

We discuss a maximum likelihood procedure for estimating parameters in possibly noncausal autoregressive processes driven by i.i.d. non-Gaussian noise. Under appropriate conditions, estimates of the parameters that are solutions to the likelihood equations exist and are asymptotically normal. The estimation procedure is illustrated with a simulation study for AR(2) processes.     

Operator-selfdecomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type

Processes of Ornstein-Uhlenbeck type on $\text{R}$^d are analogues of the Ornstein-Uhlenbeck process on $\text{R}$^d with the Brownian motion part replaced by general processes with homogeneous independent increments. The class of operator-selfdecomposable distributions of Urbanik is characterized as the class of limit distributions of such processes. Continuity of the correspondence is proved. Integro-differential equations for operator-selfdecomposable distributions are established. Examples are given for null recurrence and transience of processes of Ornstein-Uhlenbeck type on $\text{R}$¹.     

Maximum likelihood estimation of Hawkes' self-exciting point processes

Summary A maximum likelihood estimation procedure of Hawkes' self-exciting point process model is proposed with explicit presentations of the log-likelihood of the model and its gradient and Hessian. A simulation method of the process is also presented. Some numerical results are given. The Institute of Statistical Mathematics     

Non-maturing deposits modelling in a Ornstein-Uhlenbeck framework

This paper builds a multivariate Lévy-driven Ornstein-Uhlenbeck process for the management of non-maturing deposits, that are a major source of funding for banks. The contribution of the paper is both theoretical and operational. On the theoretical side, the novelty of this model is to include three independent sources of randomness in a Lévy framework: market interest rates, deposit rates and deposit volumes. The choice of a Lévy background driving process allows us to model rare but severe events. On the operational side, we propose a procedure to include severe volume outflows with positive probability in future scenarios simulation, explaining its implementation with an illustrative example using Italian banking sector data.