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On the exponential ergodicity for Lévy driven Ornstein-Uhlenbeck processes  [PDF]
Jian Wang
Mathematics , 2012,
Abstract: Based on the explicit coupling property, the ergodicity and the exponential ergodicity of L\'{e}vy driven Ornstein-Uhlenbeck processes are established.
Option pricing under stochastic volatility: the exponential Ornstein-Uhlenbeck model  [PDF]
Josep Perello,Ronnie Sircar,Jaume Masoliver
Physics , 2008, DOI: 10.1088/1742-5468/2008/06/P06010
Abstract: We study the pricing problem for a European call option when the volatility of the underlying asset is random and follows the exponential Ornstein-Uhlenbeck model. The random diffusion model proposed is a two-dimensional market process that takes a log-Brownian motion to describe price dynamics and an Ornstein-Uhlenbeck subordinated process describing the randomness of the log-volatility. We derive an approximate option price that is valid when (i) the fluctuations of the volatility are larger than its normal level, (ii) the volatility presents a slow driving force toward its normal level and, finally, (iii) the market price of risk is a linear function of the log-volatility. We study the resulting European call price and its implied volatility for a range of parameters consistent with daily Dow Jones Index data.
Multiple time scales and the exponential Ornstein-Uhlenbeck stochastic volatility model  [PDF]
Jaume Masoliver,Josep Perello
Physics , 2005, DOI: 10.1080/14697680600727547
Abstract: We study the exponential Ornstein-Uhlenbeck stochastic volatility model and observe that the model shows a multiscale behavior in the volatility autocorrelation. It also exhibits a leverage correlation and a probability profile for the stationary volatility which are consistent with market observations. All these features make the model quite appealing since it appears to be more complete than other stochastic volatility models also based on a two-dimensional diffusion. We finally present an approximate solution for the return probability density designed to capture the kurtosis and skewness effects.
Probability distribution of returns in the exponential Ornstein-Uhlenbeck model  [PDF]
Giacomo Bormetti,Valentina Cazzola,Guido Montagna,Oreste Nicrosini
Quantitative Finance , 2008, DOI: 10.1088/1742-5468/2008/11/P11013
Abstract: We analyze the problem of the analytical characterization of the probability distribution of financial returns in the exponential Ornstein-Uhlenbeck model with stochastic volatility. In this model the prices are driven by a Geometric Brownian motion, whose diffusion coefficient is expressed through an exponential function of an hidden variable Y governed by a mean-reverting process. We derive closed-form expressions for the probability distribution and its characteristic function in two limit cases. In the first one the fluctuations of Y are larger than the volatility normal level, while the second one corresponds to the assumption of a small stationary value for the variance of Y. Theoretical results are tested numerically by intensive use of Monte Carlo simulations. The effectiveness of the analytical predictions is checked via a careful analysis of the parameters involved in the numerical implementation of the Euler-Maruyama scheme and is tested on a data set of financial indexes. In particular, we discuss results for the German DAX30 and Dow Jones Euro Stoxx 50, finding a good agreement between the empirical data and the theoretical description.
Pricing of European Basket Call Option under Exponential Ornstein-Uhlenbeck Process  [PDF]
Jingwei Liu,Jiwen Luo,Xing Chen
Quantitative Finance , 2014,
Abstract: Pricing of European basket call option with n-assets and a bond is discussed in this paper, where all prices of n-assets and the bond are driven by Exponential Ornstein-Uhlenbeck processes. The close-form of European basket option pricing formula is derived. Utilizing with 1-order differential approximate numerical solution of stochastic differential equation (Milstein method), a simulation example of European basket option pricing with 3 assets is also given.
Lower estimates of transition densities and bounds on exponential ergodicity for stochastic PDE's  [PDF]
B. Goldys,B. Maslowski
Mathematics , 2004, DOI: 10.1214/009117905000000800
Abstract: A formula for the transition density of a Markov process defined by an infinite-dimensional stochastic equation is given in terms of the Ornstein--Uhlenbeck bridge and a useful lower estimate on the density is provided. As a consequence, uniform exponential ergodicity and $V$-ergodicity are proved for a large class of equations. We also provide computable bounds on the convergence rates and the spectral gap for the Markov semigroups defined by the equations. The bounds turn out to be uniform with respect to a large family of nonlinear drift coefficients. Examples of finite-dimensional stochastic equations and semilinear parabolic equations are given.
Ergodicity for a stochastic Hodgkin-Huxley model driven by Ornstein-Uhlenbeck type input  [PDF]
R. H?pfner,E. L?cherbach,M. Thieullen
Mathematics , 2013,
Abstract: We consider a model describing a neuron and the input it receives from its dendritic tree when this input is a random perturbation of a periodic deterministic signal, driven by an Ornstein-Uhlenbeck process. The neuron itself is modeled by a variant of the classical Hodgkin-Huxley model. Using the existence of an accessible point where the weak Hoermander condition holds and the fact that the coefficients of the system are analytic, we show that the system is non-degenerate. The existence of a Lyapunov function allows to deduce the existence of (at most a finite number of) extremal invariant measures for the process. As a consequence, the complexity of the system is drastically reduced in comparison with the deterministic system.
On Ornstein-Uhlenbeck driven by Ornstein-Uhlenbeck processes  [PDF]
Bernard Bercu,Frederic Proia,Nicolas Savy
Statistics , 2012,
Abstract: We investigate the asymptotic behavior of the maximum likelihood estimators of the unknown parameters of positive recurrent Ornstein-Uhlenbeck processes driven by Ornstein-Uhlenbeck processes.
Ergodicity and mixing of W*-dynamical systems in terms of joinings  [PDF]
Rocco Duvenhage
Mathematics , 2008,
Abstract: We study characterizations of ergodicity, weak mixing and strong mixing of W*-dynamical systems in terms of joinings and subsystems of such systems. Ergodic joinings and Ornstein's criterion for strong mixing are also discussed in this context.
Mixing time and local exponential ergodicity of the East-like process in $\mathbf Z^d$  [PDF]
Paul Chleboun,Alessandra Faggionato,Fabio Martinelli
Mathematics , 2015,
Abstract: The East process, a well known reversible linear chain of spins, represents the prototype of a general class of interacting particle systems with constraints modeling the dynamics of real glasses. In this paper we consider a generalization of the East process living in the d-dimensional lattice and we establish new progresses on the out- of-equilibrium behavior. In particular we prove a form of (local) exponential ergodicity when the initial distribution is far from the stationary one and we prove that the mixing time in a finite box grows linearly in the side of the box.
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