Non-standard limits for a family of autoregressive stochastic sequences

由红噪声驱动的系统的随机稳定性

On stochastic stability of systems driven by red noise

该研究针对驱动噪声为自回归红噪声的离散时间非线性随机动力系统，在标准耗散性条件下证明其总变差意义下几何快速收敛至平稳分布，丰富了随机动力系统稳定性理论。

具有类康托尔边缘分布的自动回归过程的极大半稳定极值律

Max-semistable extreme value laws for autoregressive processes with Cantor-like marginals

本文研究边缘分布类似康托尔函数的自动回归过程，证明其边缘分布属于极大半稳定分布的吸引域，并通过引入极值指数得到该过程的极值律，同时揭示其与确定性动力系统极值的联系。

On the existence and position of the farthest peaks of a family of stochastic heat and wave equations

We study a family of non-linear stochastic heat equations in (1+1) dimensions, driven by the generator of a Lévy process and space-time white noise. We assume that the underlying Lévy process has finite exponential moments in a neighborhood of the origin and that the initial condition has exponential decay at infinity. Then we prove that under natural conditions on the non-linearity: (i) The absolute moments of the solution to our stochastic heat equation grow exponentially with time; and (ii) The distances to the origin of the farthest high peaks of those moments grow exactly linearly with ti…

Stochastic modeling of stratospheric temperature

This study suggests a stochastic model for time series of daily-zonal (circumpolar) mean stratospheric temperature at a given pressure level. It can be seen as an extension of previous studies which have developed stochastic models for surface temperatures. The proposed model is a sum of a deterministic seasonality function and a Lévy-driven multidimensional Ornstein-Uhlenbeck process, which is a mean-reverting stochastic process. More specifically, the deseasonalized temperature model is an order 4 continuous time autoregressive model, meaning that the stratospheric temperature is modeled to …

Irregular Stochastic differential equations driven by a family of Markov processes

Using heat kernel estimates, we prove the pathwise uniqueness for strong solutions of irregular stochastic differential equation driven by a family of Markov process, whose generator is a non-local and non-symmetric Lévy type operator. Due to the extra term $1_{[0,σ(X_{s-},z)]}(r)$ in multiplicative noise, we need to derive some new regularity results for the generator and use a trick of mixing $L_1$ and $L_2$-estimates by Kurtz and Protter \cite{Ku-Po}.

The Family of Alpha,[a,b] Stochastic Orders: Risk vs. Expected Value

In this paper we provide a novel family of stochastic orders that generalizes second order stochastic dominance, which we call the $α,[a,b]$-concave stochastic orders. These stochastic orders are generated by a novel set of "very" concave functions where $α$ parameterizes the degree of concavity. The $α,[a,b]$-concave stochastic orders allow us to derive novel comparative statics results for important applications in economics that cannot be derived using previous stochastic orders. In particular, our comparative statics results are useful when an increase in a lottery's riskiness changes the …

A New Parameterized Family of Stochastic Particle Flow Filters

In this paper, we are interested in obtaining answers to the following questions for particle flow filters: Can we provide a theoretical guarantee that particle flow filters give correct results such as unbiased estimates? Are particle flows stable and under what conditions? Can we have one particle flow filter, rather than multiple seemingly different ones? To answer these questions, we first derive a parameterized family of stochastic particle flow filters, in which particle flows are driven by a linear combination of prior knowledge and measurement likelihood information. We then show that …

Distorted stochastic dominance: a generalized family of stochastic orders

We study a generalized family of stochastic orders, semiparametrized by a distortion function H, namely H-distorted stochastic dominance, which may determine a continuum of dominance relations from the first- to the second-order stochastic dominance (and beyond). Such a family is especially suitable for representing a decision maker's preferences in terms of risk aversion and may be used in those situations in which a strong order does not have enough discriminative power, whilst a weaker one is poorly representative of some classes of decision makers. In particular, we focus on the class of p…

Stochastic Ordering of Exponential Family Distributions and Their Mixtures

We investigate stochastic comparisons between exponential family distributions and their mixtures with respect to the usual stochastic order, the hazard rate order, the reversed hazard rate order, and the likelihood ratio order. A general theorem based on the notion of relative log-concavity is shown to unify various specific results for the Poisson, binomial, negative binomial, and gamma distributions in recent literature. By expressing a convolution of gamma distributions with arbitrary scale and shape parameters as a scale mixture of gamma distributions, we obtain comparison theorems concer…

Transition from ergodic to explosive behavior in a family of stochastic differential equations

We study a family of quadratic stochastic differential equations in the plane, motivated by applications to turbulent transport of heavy particles. Using Lyapunov functions, we find a critical parameter value $α_{1}=α_{2}$ such that when $α_{2}>α_{1}$ the system is ergodic and when $α_{2}<α_{1}$ solutions are not defined for all times. Hörmander's hypoellipticity theorem and geometric control theory are also utilized.

A family of fractional diffusion equations derived from stochastic harmonic chains with long-range interactions

We consider one-dimensional infinite chains of harmonic oscillators with stochastic perturbations and long-range interactions which have polynomial decay rate $|x|^{-θ}, x \to \infty, θ> 1$, where $x \in \mathbb{Z}$ is the interaction range. We prove that if $2< θ\le 3$, then the time evolution of the macroscopic thermal energy distribution is superdiffusive and governed by a fractional diffusion equation with exponent $\frac{3}{7-θ}$, while if $θ> 3$, then the exponent is $\frac{3}{4}$. The threshold is $ θ= 3$ because the derivative of the dispersion relation diverges as $k \to 0$ when $θ\le…

指数族可简化的随机矩阵特征化

Characterization of Exponential Families of Lumpable Stochastic Matrices

本文研究了可简化转移矩阵构成指数族的条件，提出了一种基于维度的高效验证方法。