Sort by
On the Markov-switching bilinear processes: stationarity, higher-order moments and β-mixing

This article investigates some probabilistic properties and statistical applications of general Markov-switching bilinear processes that offer remarkably rich dynamics and complex behaviour to model non-Gaussian data with structural changes. In these models, the parameters are allowed to depend on unobservable time-homogeneous and stationary Markov chain with finite state space. So, some basic issues concerning this class of models including necessary and sufficient conditions ensuring the existence of ergodic stationary (in some sense) solutions, existence of finite moments of any order and -mixing are studied. As a consequence, we observe that the local stationarity of the underlying process is neither sufficient nor necessary to obtain the global stationarity. Also, the covariance functions of the process and its power are evaluated and it is shown that the second (respectively, higher)-order structure is similar to some linear processes, and hence admit representation. We establish also sufficient conditions for the model to be mixing and geometrically ergodic. We then use these results to give sufficient conditions for mixing of a family of processes. A number of illustrative examples are given to clarify the theory and the variety of applications.

Relevant
A class of stochastic differential equations with super-linear growth and non-Lipschitz coefficients

The purpose of this paper is to study some properties of solutions to one-dimensional as well as multidimensional stochastic differential equations (SDEs in short) with super-linear growth and non-Lipschitz conditions on the coefficients. Taking inspiration from [K. Bahlali, E.H. Essaky, M. Hassani, and E. Pardoux Existence, uniqueness and stability of backward stochastic differential equation with locally monotone coefficient, C.R.A.S. Paris. 335(9) (2002), pp. 757–762; K. Bahlali, E. H. Essaky, and H. Hassani, Multidimensional BSDEs with super-linear growth coefficients: Application to degenerate systems of semilinear PDEs, C. R. Acad. Sci. Paris, Ser. I. 348 (2010), pp. 677-682; K. Bahlali, E. H. Essaky, and H. Hassani, p-Integrable solutions to multidimensional BSDEs and degenerate systems of PDEs with logarithmic nonlinearities, (2010). Available at arXiv:1007.2388v1 [math.PR]], we introduce a new local condition which ensures the pathwise uniqueness, as well as the non-contact property. We moreover show that the solution produces a stochastic flow of continuous maps and satisfies a large deviations principle of Freidlin–Wentzell type. Our conditions on the coefficients go beyond the existing ones in the literature. For instance, the coefficients are not assumed uniformly continuous and therefore cannot satisfy the classical Osgood condition. The drift coefficient could not be locally monotone and the diffusion is neither locally Lipschitz nor uniformly elliptic. Our conditions on the coefficients are, in some sense, near the best possible. Our results are sharp and mainly based on Gronwall lemma and the localization of the time parameter in concatenated intervals.

Relevant