Narrow Results

A directed network such as the WWW can be represented by a transition matrix. Comparing this matrix to a Frobenius–Perron matrix of a chaotic piecewise-linear one-dimensional map whose domain can be divided into Markov subintervals, we are able to relate the network structure itself to chaotic dynamics. Just like various large deviation properties of local expansion rates (finite-time Lyapunov exponents) related to chaotic dynamics, we can also discuss those properties of network structure.

We propose a new framework to measure the risk of a single asset and of a portfolio of financial assets which takes the agent's investment horizon into account. The methodology is based on the moderate and large deviations theory in its simplest form. We show how it can be used to select optimal portfolios given investors' planning horizons and preferences for fatter right or left tails. For practical purposes, we introduce a new parameter, the "dilation exponent" α to characterize asset returns' distributions beyond the information contained in the mean-variance framework. We estimate α for Swiss individual stocks and for MSCI country and sector stock market indices. Finally, we show how to use the dilation exponent in conjunction with Sharpe's ratio for portfolio allocation purposes.

We rigorize the work of Lewis (2007) and Durrleman (2005) on the small-time asymptotic behavior of the implied volatility under the Heston stochastic volatility model (Theorem 2.1). We apply the Gärtner-Ellis theorem from large deviations theory to the exponential affine closed-form expression for the moment generating function of the log forward price, to show that it satisfies a small-time large deviation principle. The rate function is computed as Fenchel-Legendre transform, and we show that this can actually be computed as a standard Legendre transform, which is a simple numerical root-finding exercise. We establish the corresponding result for implied volatility in Theorem 3.1, using well known bounds on the standard Normal distribution function. In Theorem 3.2 we compute the level, the slope and the curvature of the implied volatility in the small-maturity limit At-the-money, and the answer is consistent with that obtained by formal PDE methods by Lewis (2000) and probabilistic methods by Durrleman (2004).

The limiting behavior of the expectation of the super-Brownian motion with super-Brownian immigration under the quenched probability is considered: A central limit theorem is proved for d≥3, an ergodic property is considered for d =2 and a local large deviation is obtained for d = 3.

Large and moderate deviation principles are proved for the occupation time process of a subcritical branching superprocess with immigration.

We derive a large deviation principle for occupation time of super α-stable process in ℝ^d with d > 2α. The decay of tail probabilities is shown to be exponential and the rate function is characterized. Our result can be considered as a counterpart of Lee's work on large deviations for occupation times of super-Brownian motion in ℝ^d for dimension d > 4 (see Ref. 10).

In this paper large deviations for mixture of large deviation systems are investigated. Some general results are presented, with conditions either less restrictive or more practical for verification. Applications are illustrated in a concrete case-mixture of products of probability measures (homogeneous or inhomogeneous). We also discussed potential applications to problems of statistic inference for hidden Markov models.

We consider the differentiability of a spectral function generated by a Lévy process M with characteristic exponent |ξ|^αl(|ξ|²), where l(x) is a slowly varying function at ∞. As an application, we obtain the large deviation principle for positive continuous additive functionals of M. Finally, we show that the exponent l(x) = (log(1 + x))^β/2 (0 < β < 2 - α) is an example for which our theorem is applicable.

As is known, the solution of problem (1) converges to the solution of problem (3) uniformly on any finite time interval as μ ↓ 0. We study relations between large deviations for these two processes and as ε ↓ 0. A special attention is paid to the case of linear systems, when the quasi-potential is calculated explicitly.

We prove large deviation principles for solutions of small perturbations of SDEs in Hölder norms and Sobolev norms, where the SDEs have non-Markovian coefficients. As an application, we obtain a large deviation principle for solutions of anticipating SDEs in terms of (r, p) capacities on the Wiener space.

Symmetric random walks can be arranged to converge to a Wiener process in the area of normal deviation. However, random walks and Wiener processes have, in general, different asymptotics of the large deviation probabilities. The action functionals for random-walks and Wiener processes are compared in this paper. The correction term is calculated. Exit problem and stochastic resonance for random-walk-type perturbation are also considered and compared with the white-noise-type perturbation.

We study large deviation asymptotics for processes defined in terms of continued fraction digits. We use the continued fraction digit sum process to define a stopping time and derive a joint large deviation asymptotic for the upper and lower fluctuation processes. Also a large deviation asymptotic for single digits is given.

We prove the large deviation principle for continuous additive functionals under certain assumptions. The underlying symmetric Markov processes include Brownian motion and symmetric and relativistic α-stable processes.

A Freidlin–Wentzell type large deviation principle is established for stochastic partial differential equations with slow and fast time-scales, where the slow component is a one-dimensional stochastic Burgers equation with small noise and the fast component is a stochastic reaction-diffusion equation. Our approach is via the weak convergence criterion developed in [A. Budhiraja and P. Dupuis, A variational representation for positive functionals of infinite dimensional Brownian motion, Probab. Math. Statist. 20 (2000) 39–61].

Uniform probability distributions on ${\ell }_p$ balls and spheres have been studied extensively and are known to behave like product measures in high dimensions. In this note we consider the uniform distribution on the intersection of a simplex and a sphere. Certain new and interesting features, such as phase transitions and localization phenomena emerge.

We apply large deviation theory to assess the probability that a trading system performs below or above a certain threshold. Our technique does not require that the distribution of the performance criterion obeys a closed-form equation, and can accept as input empirical distributions given under the form of frequency histograms obtained by backtesting or from prior use of the trading system. A nice property of the technique is that it can be easily automated and integrated into a trading platform. Furthermore, the approach is not limited to a single trading system but can be applied on portfolio of trading systems.

Stein's method is a powerful tool in estimating accuracy of various probability approximations. It works for both independent and dependent random variables. It works for normal approximation and also for non-normal approximation. The method has been successfully applied to study the absolute error of approximations and the relative error as well. In contrast to the classical limit theorems, the self-normalized limit theorems require no moment assumptions or much less moment assumptions. This paper is devoted to the latest developments on Stein's method and self-normalized limit theory. Starting with a brief introduction on Stein's method, recent results are summarized on normal approximation for smooth functions and Berry-Esseen type bounds, Cramér type moderate deviations under a general framework of the Stein identity, non-normal approximation via exchangeable pairs, and a randomized exponential concentration inequality. For self-normalized limit theory, the focus will be on a general self-normalized moderate deviation, the self-normalized saddlepoint approximation without any moment assumption, Cramér type moderate deviations for maximum of self-normalized sums and for Studentized U-statistics. Applications to the false discovery rate in simultaneous tests as well as some open questions will also be discussed.