Fractal sets determined by dilation-stable processes

Let be a dilation-stable process on . We determine a Hausdorff measure function (a) such that the fractal set X[0,1]={X(t):0t1} has positive finite -measure. We also investigate the packing measure of X[0,1].     

Stochastic equations of super-Lévy processes with general branching mechanism

In this work, the process of distribution functions of a one-dimensional super-Lévy process with general branching mechanism is characterized as the pathwise unique solution of a stochastic integral equation driven by time–space Gaussian white noises and Poisson random measures. This generalizes the recent work of Xiong (2013), where the result for a super-Brownian motion with binary branching mechanism was obtained.     

Stochastic processes indexed by hypergroups II

Further results on weakly stationary processes indexed by hypergroups are presented. The concept of translation operators is developed; processes on orbit spaces and double coset spaces are constructed. It is shown that every weakly stationary process indexed by a hypergroupK with centerC contains a maximalK//C-weakly stationary component. New examples forK-weakly stationary processes are continuous estimates of the mean of a weakly stationary process, isotropic random fields, andK-oscillations.     

Some fractal sets determined by stable processes

Summary LetY _i be independent stable subordinators in (, ,P) with indices 0< _i <1 andR _i are the ranges ofY _i ,i=1, 2. We are able to find the exact Hausdorff measure and packing measure results for the product setsR ₁×R ₂, and whenever ₁ + ₂ 1/2, we deduce results for the vector sumR ₁R ₂={x+y:xR ₁,yR ₂}.     

Stochastic processes indexed by hypergroups. I

For many applications it is desirable to have extensions of the classical theory of weakly stationary processes to certain classes of nonstationary ones. A large family for which Fourier methods still play a major role is the harmonizable class and some related processes. The purpose of this paper is to initiate the study of a class of stochastic processes with a stationarity condition based on the notion of hypergroups.     

The asymptotic manifold of a perturbed linear differential equation as determined by the comparison principle

The asymptotic properties of the solutions of a linear homogeneous system of differential equations determine, under suitable restrictions, the asymptotic properties of a set of solutions of a nonlinear perturbation of this linear equation. The comparison principle is used here to generate an asymptotic manifold of the perturbed equation. The majorant function that is used in connection with the comparison technique is usually assumed to be nondecreasing in the dependent variable. However, properties of the asymptotic manifold are discussed here under the opposite monotonicity assumption, namely, that the majorant function is nonincreasing in the dependent variable. This type of majorant, function arises, for example, in certain gravitation problems. The main result on the structure of asymptotic manifolds which have an asymptotic uniformity is that solutions close to the manifold are either in the manifold or do not exist in the future. This research was supported in part by the National Science Foundation under grant GP-11543. Entrata in Redazione il 6 giugno 1970.     

Stochastic programming with random processes

Three possible approaches to stochastic programming problems defined in time (so that they contain random processes) are described in this paper: (1) an application of the extremal theory of random processes; (2) an exponential penalty model approach related to scenario analysis; (3) a modification of the entropic penalty approach. Explicit results are derived for some special cases.     

Stochastic integration for set-indexed processes

Diagonalizable derivations of a finite-dimensional algebra usually span an ideal in the Lie algebra of all derivations. This ideal is studied for underlying graded, monomial, and path algebras.     

Stochastic processes associated with a symmetric oscillatory poisson process

A symmetric oscillatory Poisson process is defined and its stochastic features studied. The process represented by the symbolic integral of this oscillatory Poisson process is then discussed in detail. The results obtained are applied to the well-known stochastic problem of multiple scattering of charged particles in their passage through matter.     

Large deviation principle for diffusion processes under a sublinear expectation

We represent the exponential moment of the Brownian functionals under a nonlinear expectation according to the solution to a backward stochastic differential equation.As an application,we establish a large deviation principle of the Freidlin and Wentzell type under the corresponding nonlinear probability for diffusion processes with a small diffusion coefficient.     

Moderate deviation principle for autoregressive processes

A moderate deviation principle for autoregressive processes is established. As statistical applications we provide the moderate deviation estimates of the least square and the Yule–Walker estimators of the parameter of an autoregressive process. The main assumption on the autoregressive process is the Gaussian integrability condition for the noise, which is weaker than the assumption of Logarithmic Sobolev Inequality in [H. Djellout, A. Guillin, L. Wu, Moderate deviations of empirical periodogram and nonlinear functionals of moving average processes, Ann. I. H. Poincaré-PR 42 (2006) 393–416].     

Stochastic order for alpha-permanental point processes

We establish relations of stochastic comparison among point processes elements of the set of alpha-permanental point processes. This set contains in particular, the determinantal point processes, the Poisson point processes and the permanental point processes. We show that these three classes of point processes can be ordered according to the increasing stochastic order. Elementary particles provide illustrations of some of the obtained relations of stochastic comparison.     

Stochastic averaging principle for systems with pathwise uniqueness

An ergodic type averaging assumption is used to phase out the fast mode component of a system of singularly perturbed stochastic differential equations. The weak convergence of the slow mode variables, implied by the tightness condition, is realized as almost everywhere convergence in a modified probability space. The pathwise uniqueness property is used to unify the weak limit of the slow mode with the unique solution of the reduced System     

Stochastic maximum principle for optimal control of SPDEs

In this Note, we give the stochastic maximum principle for optimal control of stochastic PDEs in the general case (when the control domain need not be convex and the diffusion coefficient can contain a control variable).     

Stochastic control processes and management science

In the stochastic control processes, the criterion which minimizes the probability that the state variables of the overall stage exceed the fixed level is probably of greater importance. Although it does not resolve the arbitrariness inherent in the choice of a criterion of performance, this is a conforming result. The paper is divided into three parts. First, the mathematical formulation, the theorem of existence and uniqueness of solution, and the optimal policy in the equations created from our stochastic control processes are considered. Second, problems involving some stochastic control processes in a fuzzy environment are considered. Third, applications of the inventory control problem, the statistical quality control processes, and a control process connected with the Van der Pol equation are presented. Finally, the future problems are discussed.