Stopped Markov chains with stationary occupation times

Summary. Let E be a finite set equipped with a group G of bijective transformations and suppose that X is an irreducible Markov chain on E that is equivariant under the action of G. In particular, if E=G with the corresponding transformations being left or right multiplication, then X is a random walk on G. We show that when X is started at a fixed point there is a stopping time U such that the distribution of the random vector of pre-U occupation times is invariant under the action of G. When G acts transitively (that is, E is a homogeneous space), any non-zero, finite expectation stopping time with this property can occur no earlier than the time S of the first return to the starting point after all states have been visited. We obtain an expression for the joint Laplace transform of the pre-S occupation times for an arbitrary finite chain and show that even for random walk on the group of integers mod r the pre-S occupation times do not generally have a group invariant distribution. This appears to contrast with the Brownian analog, as there is considerable support for the conjecture that the field of local times for Brownian motion on the circle prior to the counterpart of S is stationary under circular shifts. Received: 6 December 1995 / In revised form: 11 June 1997     

A leavable bounded-velocity stochastic control problem

This paper studies bounded-velocity control of a Brownian motion when discretionary stopping, or ‘leaving’, is allowed. The goal is to choose a control law and a stopping time in order to minimize the expected sum of a running and a termination cost, when both costs increase as a function of distance from the origin. There are two versions of this problem: the fully observed case, in which the control multiplies a known gain, and the partially observed case, in which the gain is random and unknown. Without the extra feature of stopping, the fully observed problem originates with Beneš (Stochastic Process. Appl. 2 (1974) 127–140), who showed that the optimal control takes the ‘bang–bang’ form of pushing with maximum velocity toward the origin. We show here that this same control is optimal in the case of discretionary stopping; in the case of power-law costs, we solve the variational equation for the value function and explicitly determine the optimal stopping policy.We also discuss qualitative features of the solution for more general cost structures. When no discretionary stopping is allowed, the partially observed case has been solved by Beneš et al. (Stochastics Monographs, Vol. 5, Gordon & Breach, New York and London, pp. 121–156) and Karatzas and Ocone (Stochastic Anal. Appl. 11 (1993) 569–605). When stopping is allowed, we obtain lower bounds on the optimal stopping region using stopping regions of related, fully observed problems.     

Bulk universality for Wigner matrices

We consider N × N Hermitian Wigner random matrices H where the probability density for each matrix element is given by the density ν(x) = e^?U(x). We prove that the eigenvalue statistics in the bulk are given by the Dyson sine kernel provided that U ∈ C⁶( \input amssym $\Bbb R$ ) with at most polynomially growing derivatives and ν(x) ≥ Ce^?C|x| for x large. The proof is based upon an approximate time reversal of the Dyson Brownian motion combined with the convergence of the eigenvalue density to the Wigner semicircle law on short scales. © 2010 Wiley Periodicals, Inc.     

Conditional Essential Suprema with Applications

The conditional supremum of a random variable X on a probability space given a sub--algebra is defined and proved to exist as an application of the Radon–Nikodym theorem in L ^\infty. After developing some of its properties we use it to prove a new ergodic theorem showing that a time maximum is a space maximum. The concept of a maxingale is introduced and used to develop the new theory of optimal stopping in L ^\infty and the concept of an absolutely optimal stopping time. Finally, the conditional max is used to reformulate the optimal control of the worst-case value function.     

Hitting Time Problems of Sticky Brownian Motion and Their Applications in Optimal Stopping and Bond Pricing

This paper investigates the hitting time problems of sticky Brownian motion and their applications in optimal stopping and bond pricing. We study the Laplace transform of first hitting time over the constant and random jump boundary, respectively. The results about hitting the constant boundary serve for solving the optimal stopping problem of sticky Brownian motion. By introducing the sharpo ratio, we settle the bond pricing problem under sticky Brownian motion as well. An interesting result shows that the sticky point is in the continuation region and all the results we get are in closed form.

     

Fractional Generalized Random Fields of Variable Order

Abstract

We study the class of random fields having their reproducing kernel Hilbert space isomorphic to a fractional Sobolev space of variable order on ? ⁿ . Prototypes of this class include multifractional Brownian motion, multifractional free Markov fields, and multifractional Riesz–Bessel motion. The study is carried out using the theory of generalized random fields defined on fractional Sobolev spaces of variable order. Specifically, we consider the class of generalized random fields satisfying a pseudoduality condition of variable order. The factorization of the covariance operator of the pseudodual allows the definition of a white-noise linear filter representation of variable order. In the ordinary case, the Hölder continuity, in the mean-square sense, of the class of random fields introduced is proved, and its mean-square Hölder spectrum is defined in terms of the variable regularity order of the functions in the associated reproducing kernel Hilbert space. The pseudodifferential representation of variable order of the resulting class of multifractal random fields is also defined. Some examples of pseudodifferential models of variable order are then given.     

Minimal cyclic random motion in and hyper-Bessel functions

We obtain the explicit distribution of the position of a particle performing a cyclic, minimal, random motion with constant velocity c in . The n+1 possible directions of motion as well as the support of the distribution form a regular hyperpolyhedron (the first one having constant sides and the other expanding with time t), the geometrical features of which are here investigated.The distribution is obtained by using order statistics and is expressed in terms of hyper-Bessel functions of order n+1. These distributions are proved to be connected with (n+1)th order p.d.e. which can be reduced to Bessel equations of higher order.Some properties of the distributions obtained are examined. This research has been inspired by a conjecture formulated in Orsingher and Sommella [E. Orsingher, A.M. Sommella, A cyclic random motion in R³ with four directions and finite velocity, Stochastics Stochastics Rep. 76 (2) (2004) 113–133] which is here proved to be false.     

Asymptotic Behavior of the Perturbed Empirical Distribution-Functions Evaluated at a Random Point for Absolutely Regular Sequences

Let _n be an estimator obtained by integrating a kernel type density estimator based on a random sample of size n from smooth distribution function F. A central limit theorem is established for the target statistic _n(U_n) where the underlying random variable form an absolutely regular stationary process and where {U_n} is a sequence of U-statistics. The result obtained generalizes Puri and Ralescu (1986, J. Multivariate Anal.19, 273-279) under the iid set-up.     

Homogenization of a class of one-dimensional nonconvex viscous Hamilton-Jacobi equations with random potential

Abstract

We prove the homogenization of a class of one-dimensional viscous Hamilton-Jacobi equations with random Hamiltonians that are nonconvex in the gradient variable. Due to the special form of the Hamiltonians, the solutions of these PDEs with linear initial conditions have representations involving exponential expectations of controlled Brownian motion in a random potential. The effective Hamiltonian is the asymptotic rate of growth of these exponential expectations as time goes to infinity and is explicit in terms of the tilted free energy of (uncontrolled) Brownian motion in a random potential. The proof involves large deviations, construction of correctors which lead to exponential martingales, and identification of asymptotically optimal policies.     

Tanaka Formula for Multidimensional Brownian Motions

We study the Tanaka formula for multidimensional Brownian motions in the framework of generalized Wiener functionals. More precisely, we show that the submartingale U(B _t –x) is decomposed in the sence of generalized Wiener functionals into the sum of a martingale and the Brownian local time, U being twice of the kernel of Newtonian potential and B _t the multidimensional Brownian motion. We also discuss on an aspect of the Tanaka formula for multidimensional Brownian motions as the Doob–Meyer decomposition.     

Dispersing hash functions

We define a family of functions F from a domain U to a range R to be dispersing if for every set S ? U of a certain size and random h ∈ F, the expected value of ∣S∣ – ∣h[S]∣ is not much larger than the expectation if h had been chosen at random from the set of all functions from U to R. We give near‐optimal upper and lower bounds on the size of dispersing families and present several applications where using such a family can reduce the use of random bits compared to previous randomized algorithms. A close relationship between dispersing families and extractors is exhibited. This relationship provides good explicit constructions of dispersing hash functions for some parameters, but in general the explicit construction is left open. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Estimation of the Failure Rate in a Partially Accelerated Life Test: A Sequential Approach

Abstract

Consider the situation in which a group of units are put on a partially accelerated life test. It is assumed that the lifelengths of the units are independent and exponentially distributed random variables with common failure rate θ, and that θ is the value of a random variable having a gamma distribution. A two‐stage sequential procedure for estimating θ under the squared error loss is proposed. In the first stage, the units are put on the test under normal stress up to time t, where t is determined as a stopping time that minimizes the expected loss plus cost of running the test. In the second stage, the stress is raised to a higher level for those units that did not fail by time t and held constant until they all fail. The accumulated data are then used to estimate θ with the Bayes estimator.     

Approach to equilibrium of a rotating sphere in a Stokes flow

We study the unsteady rotary motion of a sphere immersed in a Stokes fluid. The equation of motion for the sphere leads to an integro-differential equation, and we are interested in the asymptotic behavior in time of the solution. Preparing initially the system (sphere + fluid) as a stationary state, we prove that the angular velocity of the sphere slows down with a law t ^−3/2 if no other forces than the one exerted by the fluid act on the sphere, while if the sphere is subject also to an elastic torque the asymptotic behavior of the angular position of the sphere is t ^−γ , with γ = 5/2 if the initial angular velocity is zero, γ = 3/2 otherwise. This behavior is due to the memory effect of the surrounding fluid. We discuss briefly other initial preparations of the system.     

Mixing of the Glauber dynamics for the ferromagnetic Potts model

We present several results on the mixing time of the Glauber dynamics for sampling from the Gibbs distribution in the ferromagnetic Potts model. At a fixed temperature and interaction strength, we study the interplay between the maximum degree (Δ) of the underlying graph and the number of colours or spins (q) in determining whether the dynamics mixes rapidly or not. We find a lower bound L on the number of colours such that Glauber dynamics is rapidly mixing if at least L colours are used. We give a closely‐matching upper bound U on the number of colours such that with probability that tends to 1, the Glauber dynamics mixes slowly on random Δ‐regular graphs when at most U colours are used. We show that our bounds can be improved if we restrict attention to certain types of graphs of maximum degree Δ, e.g. toroidal grids for Δ = 4. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 48, 21–52, 2016     

Rate of growth of the coalescent set in a coalescing stochastic flow

Consider an isotropic stochastic flow in R_d (i.e. a simultaneous random, correlated motion of all points in space), where d=l,2 or 3, such that the joint law of the motion of two particles allows the particles to meet and coalesce in finite time. The coalescent set J _t is a random subset of R_d consisting of the initial positions of particles which have coalesced by time t with the particle which started at 0. We show that the expected volume of J t grows at a rate proportional to when d=1, and at rates close to proportional to t/log t (resp. t) when d = 2 (resp. d=3). We give an example of a coalescing stochastic flow when d = 3. These results are analogous to growth rates of expected population size of a surviving type in the "invasion process" described by Clifford and Sudbury     

Ruin probability and joint distributions of some actuarial random vectors in the compound pascal model

The compound negative binomial model,introduced in this paper,is a discrete time version.We discuss the Markov properties of the surplus process,and study the ruin probability and the joint distributions of actuarial random vectors in this model.By the strong Markov property and the mass function of a defective renewal sequence,we obtain the explicit expressions of the ruin probability,the finite-horizon ruin probability,the joint distributions of T,U(T-1),|U(T)| and 0≤inn相似文献   

Current Fluctuations for Independent Random Walks in Multiple Dimensions

Consider a system of particles evolving as independent and identically distributed (i.i.d.) random walks. Initial fluctuations in the particle density get translated over time with velocity [(v)\vec]\vec{v}, the common mean velocity of the random walks. Consider a box centered around an observer who starts at the origin and moves with constant velocity [(v)\vec]\vec{v}. To observe interesting fluctuations beyond the translation of initial density fluctuations, we measure the net flux of particles over time into this moving box. We call this the “box-current” process.     

Critical thresholds in the semiclassical limit of 2-D rotational Schrödinger equations

We consider a two-dimensional convection model augmented with the rotational Coriolis forcing, centrifugal forcing as well as the quadratic potential with a fixed Ω > 0 being the rotational frequency. This model arises in the semiclassical limit of the Gross–Pitaevskii equation for Bose–Einstein condensates in a rotational frame. We investigate whether the action of dispersive rotational forcing complemented with the underlying potential prevents the generic finite time breakdown of the free nonlinear convection. We show that the rotating equations admit global smooth solutions for and only for a subset of generic initial configurations. Thus, the global regularity depends on whether the initial configuration crosses an intrinsic critical threshold, which is quantified in terms of the initial spectral gap associated with the 2 × 2 initial velocity gradient, λ ₂ (0) − λ ₁ (0), λ _j (0)=λ _j (∇ _x U₀) as well as the initial divergence, div_x (U₀). We also prove that for the case of isotropic trapping potential the smooth velocity field is periodic if and only if the ratio of the rotational frequency and the potential frequency is a rational number. The critical thresholds are also established for the case of repulsive potential. Finally the position density and the velocity field are explicitly recorded along the deformed flow map. Received: November 12, 2003; revised: May 4, 2004     

Weak invariance principle for finite-populationU-statistics

Under the weakest possible conditions, we establish the weak invariance principle for finite-populationU-statistics in this paper. It is worth while to point out that, for the sampling without replacement, the sequence of random delements inC[0, 1], associated with the sample partial sums or theU-statistics, converges in law to the standard Brown bridge, but not to the Brown motion as in the usual case of replacement sampling.     

Orthogonal random vectors and the Hurwitz-Radon-Eckmann theorem

In several different aspects of algebra and topology the following problem is of interest: find the maximal number of unitary antisymmetric operatorsU _i inH = ℝ ⁿ with the propertyU _i U _j = −U _j U _i (i≠j). The solution of this problem is given by the Hurwitz-Radon-Eckmann formula. We generalize this formula in two directions: all the operatorsU _i must commute with a given arbitrary self-adjoint operator andH can be infinite-dimensional. Our second main result deals with the conditions for almost sure orthogonality of two random vectors taking values in a finite or infinite-dimensional Hilbert spaceH. Finally, both results are used to get the formula for the maximal number of pairwise almost surely orthogonal random vectors inH with the same covariance operator and each pair having a linear support inH⊕H. The paper is based on the results obtained jointly with N.P. Kandelaki (see [1,2,3]).