Large deviations of realized volatility

This paper studies large and moderate deviation properties of a realized volatility statistic of high frequency financial data. We establish a large deviation principle for the realized volatility when the number of high frequency observations in a fixed time interval increases to infinity. Our large deviation result can be used to evaluate tail probabilities of the realized volatility. We also derive a moderate deviation rate function for a standardized realized volatility statistic. The moderate deviation result is useful for assessing the validity of normal approximations based on the central limit theorem. In particular, it clarifies that there exists a trade-off between the accuracy of the normal approximations and the path regularity of an underlying volatility process. Our large and moderate deviation results complement the existing asymptotic theory on high frequency data. In addition, the paper contributes to the literature of large deviation theory in that the theory is extended to a high frequency data environment.     

Moderate Deviation for the Rightmost Position in a Branching Brownian Motion

We study the moderate deviation probability of the position of the rightmost particle in a branching Brownian motion and obtain its moderate deviation function. Firstly, Chauvin and Rouault studied the large deviation probability for the rightmost position in a branching Brownian motion. Recently, Derrida and Shiconsidered lower deviation for the same model. By contrast, Our main result is more extensive.     

无穷维自回归过程的中偏差原理

本文考虑无穷维自回归过程经验协方差函数的中偏差原理,仅对自回归过程的随机扰动项做了高斯可积性的假设,这个条件比[4]中的对数Sobolev不等式要弱很多.主要利用了m-相依随机变量的中偏差结果和Ellis-Grtner定理,推广了[6]的结果.     

Large deviations for the Fleming–Viot process with neutral mutation and selection,II

Large deviation principles are established for the Fleming–Viot process with neutral mutation and with selection, and the associated equilibrium measures as the sampling rate approaches zero and when the state space is equipped with the weak topology. The path-level large deviation results improve the results of Dawson and Feng (1998, Stochastic Process. Appl. 77, 207–232) in three aspects: the state space is more natural, the initial condition is relaxed, and a large deviation principle is established for the Fleming–Viot process with selection. These improvements are achieved through a detailed study of the behaviour near the boundary of the Fleming–Viot process with finite types.     

Asymptotic behavior and the moderate deviation principle for the maximum of a Dyck path

In this work, we obtain a large and moderate deviation principle for the law of the maximum of a random Dyck path. Our result extends the results of Chung (1976), Kennedy (1976) and Khorunzhiy and Marckert (2009).     

Moderate deviations for martingale differences and applications to φ -mixing sequences

For a R d -valued sequence of martingale differences { m k } k S 1 , we obtain a moderate deviation principle for the sequence of partial sums { Z n ( t ) 1 ~ k =1 [ nt ] m k / b n , t ] [0,1]}, in the space of càdlàg functions equipped with the Skorohod topology, under the following conditions: a Chen-Ledoux type condition, an exponential convergence in probability of the associated quadratic variation process of the martingale, and a condition of "Lindeberg" type. For the small jumps of Z n (·), we apply the general result of Puhalskii [Puhalskii, A. (1994). "Large deviations of semimartingales via convergence of the predictable characteristics". Stoch. Stoch. Rep. , 49 , pp. 27-85]. Following the method of Ledoux [Ledoux, M. (1992). "Sur les déviations modérées des sommes de variables aléatoires vectorielles indépendantes de même loi". Ann. Inst. H. Poincaré , 28 , pp. 267-280] and Arcones [Arcones, A. (1999). "The large deviation principle for stochastic processes", Submitted for publication], we prove that the large jumps part of Z n (·) is negligible in the sense of the moderate deviations. One can regard our result as an extension to martingale differences, of the beautiful characterization of moderate deviations for i.i.d.r.v. case due to Chen [Chen, X. (1991). "The moderate deviations of independent vectors in Banach space". Chin. J. Appl. Probab. Stat. , 7 , pp. 124-32] and Ledoux [Ledoux, M. (1992). "Sur les déviations modérées des sommes de variables aléatoires vectorielles indépendantes de même loi". Ann. Inst. H. Poincaré , 28 , pp. 267-280]. Using the Gordin [Gordin, M.I. (1969). "The central limit theorem for stationary processes". Soviet Math. Dokl. , 10 , pp. 1174-1176] decomposition, the martingale result is applied to prove the moderate deviation principle for a wide class of stationary { -mixing sequences of random variables.     

The effect of memory on functional large deviations of infinite moving average processes

The large deviations of an infinite moving average process with exponentially light tails are very similar to those of an i.i.d. sequence as long as the coefficients decay fast enough. If they do not, the large deviations change dramatically. We study this phenomenon in the context of functional large, moderate and huge deviation principles.     

Classes of sequences of real numbers, games and selection properties

This is a continuation of our investigation of classes of sequences of positive real numbers satisfying some selection principles as well as having certain game-theoretic properties. We improve main results from [D. Djur?i?, Lj.D.R. Ko?inac, M.R. ?i?ovi?, Some properties of rapidly varying sequences, J. Math. Anal. Appl. 327 (2007) 1297-1306] and [D. Djur?i?, Lj.D.R. Ko?inac, M.R. ?i?ovi?, Rapidly varying sequences and rapid convergence, Topology Appl. (2008), doi: 10.1016/j.topol.2007.05.026, in press].     

Large deviations and phase transition for random walks in random nonnegative potentials

We establish large deviation principles and phase transition results for both quenched and annealed settings of nearest-neighbor random walks with constant drift in random nonnegative potentials on Z^d${\mathbb{Z}}^d$. We complement the analysis of M.P.W. Zerner [Directional decay of the Green’s function for a random nonnegative potential on Z^d${\mathbb{Z}}^d$, Ann. Appl. Probab. 8 (1996) 246–280], where a shape theorem on the Lyapunov functions and a large deviation principle in absence of the drift are achieved for the quenched setting.     

A large deviation principle form-variate von Mises-statistics and U-statistics

A large deviation principle form-variate von Mises-statistics and U-statistics with a kernel function satisfying natural moment conditions is proved. Sanov's large deviation result for the empirical distribution function and two fundamental conservation principles in large deviation theory are the main tools. The rate functions are “drawback”-entropy functionals.     

Large deviations for Glauber dynamics of continuous gas

This paper is devoted to the large deviation principles of the Glauber-type dynamics of finite or infinite volume continuous particle systems.We prove that the level-2 empirical process satisfies the large deviation principles in the weak convergence topology,while it does not satisfy the large deviation principles in the T-topology.     

Large and Moderate Deviations for Random Walks on Nilpotent Groups

We prove large and moderate deviation estimates for products of i.i.d. r.v.'s taking values on simply connected nilpotent Lie groups as a consequence of large and moderate deviation results for stochastic processes which are solutions of O.D.E. with random coefficients.     

Moderate deviation for super-Brownian motion with immigration

We prove a moderate deviation principle for a super-Brownian motion with im-migration of all dimensions, and consequently fill the gap between the central limit theoremand large deviation principle.     

Large deviation principle of stochastic differential equations with non-Lipschitzian coefficients

We study the large deviation principle of stochastic differential equations with non-Lipschitzian and non-homogeneous coefficients. We consider at first the large deviation principle when the coefficients σ and b are bounded, then we generalize the conclusion to unbounded case by using bounded approximation program. Our results are generalization of S. Fang-T. Zhang＇s results.     

Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings

In this paper, the famous Banach contraction principle and Caristi's fixed point theorem are generalized to the case of multi-valued mappings. Our results are extensions of the well-known Nadler's fixed point theorem [S.B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969) 475-487], as well as of some Caristi type theorems for multi-valued operators, see [N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl. 141 (1989) 177-188; J.P. Aubin, Optima and Equilibria. An Introduction to Nonlinear Analysis, Grad. Texts in Math., Springer-Verlag, Berlin, 1998, p. 17; S.S. Zhang, Q. Luo, Set-valued Caristi fixed point theorem and Ekeland's variational principle, Appl. Math. Mech. 10 (2) (1989) 111-113 (in Chinese), English translation: Appl. Math. Mech. (English Ed.) 10 (2) (1989) 119-121], etc.     

LARGE DEVIATIONS FOR INFINITE DIMENSIONAL AND REVERSIBLE REACTION－DIFFUSION PROCESSES

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非线性差分方程的多重周期解

用变分方法得到一类非线性差分方程多重周期解的存在性．我们的结果推广了Cai，Yu和Guo[Comput．Math．Appl．，52（2006），1630-1647]的结果，并且这里给出的证明显著地简化了．     

On Large Deviation Convergence of Invariant Measures

We present new results on the connection between large deviation principles for trajectories of stochastic processes and the associated invariant measures. Applications to diffusion and queuing processes are provided.     

Existence of multiple periodic solutions for an SIR model with seasonality

We study an SIR model with a seasonal contact rate and a staged treatment strategy, which is an extension of our previous work [Z. Bai, Y. Zhou, Existence of two periodic solutions for a non-autonomous SIR epidemic model, Appl. Math. Model. 35 (2011) 382-391]. It is proved that the persistence and extinction of the disease are determined by the basic reproductive number (R₀) and a threshold parameter (R_c). It is obtained that the model exhibits two different bistable behaviors under certain conditions: the stable disease-free state coexists with a stable endemic periodic solution, and three endemic periodic solutions coexist with two of them being stable. Numerical simulations are presented to illustrate theoretical results.     

带移民超布朗运动的占位时中偏差

本文证明了当底空间维数d≥3时,一类带移民超布朗运动占位时过程的中偏差,其移民由Lebesgue 测度控制.可以清楚地看出,中偏差的规范化因子和速度函数恰好介于中心极限定理和大偏差之间,在 这个意义下,中偏差填补了中心极限定理和大偏差之间的空白.