Minimum contrast estimator for fractional Ornstein-Uhlenbeck processes

This paper proposes a minimum contrast methodology to estimate the drift parameter for the Ornstein-Uhlenbeck process driven by fractional Brownian motion of Hurst index, which is greater than one half. Both the strong consistency and the asymptotic normality of this minimum contrast estimator are studied based on the Laplace transform. The numerical simulation results confirm the theoretical analysis and show that the minimum contrast technique is effective and efficient.     

Pseudo almost automorphic solution to stochastic differential equation driven by Lévy process

Almost automorphic is a particular case of the recurrent motion, which has been studied in differential equations for a long time. We introduce square-mean pseudo almost automorphic and some of its properties, and then study the pseudo almost automorphic solution in the distribution sense to stochastic differential equation driven by Lévy process.     

Precise Asymptotics for Lévy Processes

Let {X（t）, t ≥ 0} be a Lévy process with EX（1） = 0 and EX^2（1） 〈 ∞. In this paper, we shall give two precise asymptotic theorems for {X（t）, t 〉 0}. By the way, we prove the corresponding conclusions for strictly stable processes and a general precise asymptotic proposition for sums of i.i.d. random variables.     

Parameter Estimation for the Discretely Observed Vasicek Model with Small Fractional Lévy Noise

The statistical inference of the Vasicek model driven by small Levy process has a long history.In this paper,we consider the problem of parameter estimation for Vasicek model dX_t=(μ-θX_t)dt+εdL_t^d,t∈[0,1],X_0=x_0,driven by small fractional Lévy noise with the known parameter d less than one half,based on discrete high-frequency observations at regularly spaced time points{t_i=i/n,i=1,2,...,n}.For the general case and the null recurrent case,the consistency as well as the asymptotic behavior of least squares estimation of unknown parametersμandθhave been established as small dispersion coefficientε→0 and large sample size n→∞simultaneously.     

A Wiener-Hopf factorization related potential measure for spectrally negative Lévy process

For spectrally negative Lévy process (SNLP), we find an expression, in terms of scale functions, for a potential measure involving the maximum and the last time of reaching the maximum up to a draw-down time. As applications, we obtain a potential measure for the reflected SNLP and recover a joint Laplace transform for the Wiener-Hopf factorization for SNLP.     

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.     

Erratum to: least squares estimation for Ornstein-Uhlenbeck processes driven by the weighted fractional brownian motion (Acta Mathematica Scientia 2016,36B(2):394-408)

We give a correction of Theorem 2.2 of Shen, Yin and Yan (2016).     

Fragmentation associated with Lévy processes using snake

We consider the height process of a Lévy process with no negative jumps, and its associated continuous tree representation. Using Lévy snake tools developed by Le Gall-Le Jan and Duquesne-Le Gall, with an underlying Poisson process, we construct a fragmentation process, which in the stable case corresponds to the self-similar fragmentation described by Miermont. For the general fragmentation process we compute a family of dislocation measures as well as the law of the size of a tagged fragment. We also give a special Markov property for the snake which is of its own interest.      

A risk model driven by Lévy processes

We present a general risk model where the aggregate claims, as well as the premium function, evolve by jumps. This is achieved by incorporating a Lévy process into the model. This seeks to account for the discrete nature of claims and asset prices. We give several explicit examples of Lévy processes that can be used to drive a risk model. This allows us to incorporate aggregate claims and premium fluctuations in the same process. We discuss important features of such processes and their relevance to risk modeling. We also extend classical results on ruin probabilities to this model. Copyright © 2003 John Wiley & Sons, Ltd.     

Approximation of quantum Lévy processes by quantum random walks

Every quantum Lévy process with a bounded stochastic generator is shown to arise as a strong limit of a family of suitably scaled quantum random walks.     

Fractional Lévy processes on Gel’fand triple and stochastic integration

In this paper, we investigate the long-range dependence of fractional Lévy processes on Gel’fand triple and construct stochastic integral with respect to fractional Lévy processes for a class of deterministic integrands.      

Product and moment formulas for iterated stochastic integrals (associated with Lévy processes)

In this paper, we obtain explicit product and moment formulas for products of iterated integrals generated by families of square integrable martingales associated with an arbitrary Lévy process. We propose a new approach applying the theory of compensated-covariation stable families of martingales. Our main tool is a representation formula for products of elements of a compensated-covariation stable family, which enables us to consider Lévy processes, with both jumps and Gaussian part.     

On the rate of convergence of weak Euler approximation for nondegenerate SDEs driven by Lévy processes

The paper studies the rate of convergence of the weak Euler approximation for solutions to SDEs driven by Lévy processes, with Hölder-continuous coefficients. It investigates the dependence of the rate on the regularity of coefficients and driving processes. The equation considered has a nondegenerate main part driven by a spherically symmetric stable process.     

On the bailout dividend problem with periodic dividend payments for spectrally negative Markov additive processes

This paper studies the bailout optimal dividend problem with regime switching under the constraint that dividend payments can be made only at the arrival times of an independent Poisson process while capital can be injected continuously in time. We show the optimality of the regime-modulated Parisian-classical reflection strategy when the underlying risk model follows a general spectrally negative Markov additive process. In order to verify the optimality, first we study an auxiliary problem driven by a single spectrally negative Lévy process with a final payoff at an exponential terminal time and characterize the optimal dividend strategy. Then, we use the dynamic programming principle to transform the global regime-switching problem into an equivalent local optimization problem with a final payoff up to the first regime switching time. The optimality of the regime modulated Parisian-classical barrier strategy can be proven by using the results from the auxiliary problem and approximations via recursive iterations.     

Stochastic control of SDEs associated with Lévy generators and application to financial optimization

This paper is concerned with the optimal control of jump type stochastic differential equations associated with (general) Lévy generators. The maximum principle is formulated for the solutions of the equations, which is inspired by N. C. Framstad, B. Øsendal and A. Sulem [J. Optim. Theory Appl., 2004, 121: 77―98] (and a continuation, J. Bennett and J. -L. Wu [Front. Math. China, 2007, 2(4): 539―558]). The result is then applied to optimization problems in financial models driven by Lévy-type processes.     

Asymptotic properties of a maximum likelihood estimator with data from a Gaussian process

We consider an estimation problem with observations from a Gaussian process. The problem arises from a stochastic process modeling of computer experiments proposed recently by Sacks, Schiller, and Welch. By establishing various representations and approximations to the corresponding log-likelihood function, we show that the maximum likelihood estimator of the identifiable parameter θσ² is strongly consistent and converges weakly (when normalized by √n) to a normal random variable, whose variance does not depend on the selection of sample points. Some extensions to regression models are also obtained.     

Weak unimodality of finite measures, and an application to potential theory of additive Lévy processes

A probability measure on is called weakly unimodal if there exists a constant such that for all 0$">,

(0.1)

Here, denotes the -ball centered at with radius 0$">.

In this note, we derive a sufficient condition for weak unimodality of a measure on the Borel subsets of . In particular, we use this to prove that every symmetric infinitely divisible distribution is weakly unimodal. This result is then applied to improve some recent results of the authors on capacities and level sets of additive Lévy processes.