Medium‐term horizon volatility forecasting: A comparative study

In this paper, volatility is estimated and then forecast using unobserved components‐realized volatility (UC‐RV) models as well as constant volatility and GARCH models. With the objective of forecasting medium‐term horizon volatility, various prediction methods are employed: multi‐period prediction, variable sampling intervals and scaling. The optimality of these methods is compared in terms of their forecasting performance. To this end, several UC‐RV models are presented and then calibrated using the Kalman filter. Validation is based on the standard errors on the parameter estimates and a comparison with other models employed in the literature such as constant volatility and GARCH models. Although we have volatility forecasting for the computation of Value‐at‐Risk in mind the methodology presented has wider applications. This investigation into practical volatility forecasting complements the substantial body of work on realized volatility‐based modelling in business. Copyright © 2007 John Wiley & Sons, Ltd.     

A new way of investigating the asymptotic behaviour of a stochastic SIS system with multiplicative noise

In this paper, we investigate a nonlinear stochastic SIS epidemic system with multiplicative noise. First, we transform the Itô’s integral into an equivalent Stratonovich integral. Then, by using the solution of Langevin equation and Ornstein–Uhlenbeck process, we prove that the system generates a random dynamical system which has a tempered compact random absorbing set, implying the condition for the extinction of the disease. Finally, the discussion and numerical simulation are given to demonstrate the obtained result.     

Exponential functionals of Lévy processes and variable annuity guaranteed benefits

Exponential functionals of Brownian motion have been extensively studied in financial and insurance mathematics due to their broad applications, for example, in the pricing of Asian options. The Black–Scholes model is appealing because of mathematical tractability, yet empirical evidence shows that geometric Brownian motion does not adequately capture features of market equity returns. One popular alternative for modeling equity returns consists in replacing the geometric Brownian motion by an exponential of a Lévy process. In this paper we use this latter model to study variable annuity guaranteed benefits and to compute explicitly the distribution of certain exponential functionals.     

Cox Point Processes Driven by Ornstein–Uhlenbeck Type Processes

The paper is devoted to the development of Cox point processes driven by nonnegative processes of Ornstein–Uhlenbeck (OU) type. Starting with multivariate temporal processes we develop formula for the cross pair correlation function. Further filtering problem is studied by means of two different approaches, either with discretization in time or through the point process densities with respect to the Poisson process. The first approach is described mainly analytically while in the second case we obtain numerical solution by means of MCMC. The Metropolis–Hastings birth–death chain for filtering can be also used when estimating the parameters of the model. In the second part we try to develop spatial and spatio-temporal Cox point processes driven by a stationary OU process. The generating functional of the point process is derived which enables evaluation of basic characteristics. Finally a simulation algorithm is given and applied.      

Multilayer Markov Chains with Applications to Polymers in Shear Flow

We present an analysis of multilayer Markov chains and apply the results to a model of a tethered polymer chain in shear flow. We find that the stationary probability measure in the direction of the flow is nonmonotonic, and has several maxima and minima for sufficiently high shear rates. This is in agreement with the experimental observation of cyclic dynamics for such polymer systems. Estimates for the stationary variance and expectation value were obtained and showed to be in accordance with our numerical results.     

Parabolic bundles on algebraic surfaces I — The Donaldson-Uhlenbeck compactification

The aim of this paper is to construct the parabolic version of the Donaldson-Uhlenbeck compactification for the moduli space of parabolic stable bundles on an algebraic surface with parabolic structures along a divisor with normal crossing singularities. We prove the non-emptiness of the moduli space of parabolic stable bundles of rank 2.     

Phylogenetic Ornstein–Uhlenbeck regression curves

Regression curves for studying trait relationships are developed herein. The adaptive evolution model is considered as an Ornstein–Uhlenbeck system whose parameters are estimated by a novel engagement of generalized least-squares and optimization. Our algorithm is implemented to ecological data.     

Volterra-type Ornstein–Uhlenbeck processes in space and time

We propose a novel class of temporo-spatial Ornstein–Uhlenbeck processes as solutions to Lévy-driven Volterra equations with additive noise and multiplicative drift. After formulating conditions for the existence and uniqueness of solutions, we derive an explicit solution formula and discuss distributional properties such as stationarity, second-order structure and short versus long memory. Furthermore, we analyze in detail the path properties of the solution process. In particular, we introduce different notions of càdlàg paths in space and time and establish conditions for the existence of versions with these regularity properties. The theoretical results are accompanied by illustrative examples.     

Maximum likelihood estimation for multiscale Ornstein–Uhlenbeck processes

We study the problem of estimating the parameters of an Ornstein–Uhlenbeck (OU) process that is the coarse-grained limit of a multiscale system of OU processes, given data from the multiscale system. We consider both the averaging and homogenization cases and both drift and diffusion coefficients. By restricting ourselves to the OU system, we are able to substantially improve the results with strong modes of convergence, and provide some intuition of what to expect in the general case. In particular, in the homogenisation case we derive optimal rates of sub-sampling to minimize the estimation errors.