On a new set-valued stochastic integral with respect to semimartingales and its applications

We present a new approach to a concept of a set-valued stochastic integral with respect to semimartingales. Such an integral, called set-valued stochastic up-trajectory integral, is compatible with the decomposition of the semimartingale. Some properties of this integral are stated. We show applicability of the new integral in set-valued stochastic integral equations driven by multidimensional semimartingales. The uniqueness theorem is presented. Then we extend the notion of the set-valued stochastic up-trajectory integral to definition of a fuzzy stochastic up-trajectory integral with respect to semimartingales. A result on uniqueness of a solution to fuzzy stochastic integral equations incorporating the new fuzzy stochastic up-trajectory integral driven by the multidimensional semimartingale is stated.     

On the Continuity and Absolute Continuity of Random Closed Sets

Abstract

In the case of real-valued random variables, the concept of absolute continuity is well-defined in terms of the absolute continuity of the probability law of a random variable with respect to the usual Lebesgue measure, since both are acting on the same Borel sigma algebra on the real line. Naturally, the same extends to random vectors with real components. A satisfactory and commonly accepted definition of absolute continuity of random closed sets is not available, while in various applications this would help in clarifying the kind of randomness of a random set. We introduce here a definition that is shown to be an extension of the concept related to real-valued random variables, such that also for random sets it is true that absolute continuity implies continuity. Significant examples and counter examples are presented to illustrate the role of our definition in concrete cases. The relationship between our definition and others in well-accepted literature is shown.     

Uniform laws of large numbers for set-valued mappings and subdifferentials of random functions

We derive a uniform (strong) Law of Large Numbers (LLN) for random set-valued mappings. The result can be viewed as an extension of both, a uniform LLN for random functions and LLN for random sets. We apply the established results to a consistency analysis of stationary points of sample average approximations of nonsmooth stochastic programs.     

Optional and predictable projections of set-valued measurable processes

Abstract. In this paper,the optional and predictable projections of set-valued measurable pro-cesses are studied. The existence and uniqueness of optional and predictable projections of set-valued measurable processes are proved under proper circumstances.     

From the Ehrenfest model to time-fractional stochastic processes

The Ehrenfest model is considered as a good example of a Markov chain. I prove in this paper that the time-fractional diffusion process with drift towards the origin, is a natural generalization of the modified Ehrenfest model. The corresponding equation of evolution is a linear partial pseudo-differential equation with fractional derivatives in time, the orders lying between 0 and 1. I focus on finding a precise explicit analytical solution to this equation depending on the interval of the time. The stationary solution of this model is also analytically and numerically calculated. Then I prove that the difference between the discrete approximate solution at time t_n, n≥0, and the stationary solution obeys a power law with exponent between 0 and 1. The reversibility property is discussed for the Ehrenfest model and its fractional version with a new observation.     

On solutions of stochastic differential equations with parameters modeled by random sets

We consider ordinary stochastic differential equations whose coefficients depend on parameters. After giving conditions under which the solution processes continuously depend on the parameters random compact sets are used to model the parameter uncertainty. This leads to continuous set-valued stochastic processes whose properties are investigated. Furthermore, we define analogues of first entrance times for set-valued processes called first entrance and inclusion times. The theoretical concept is applied to a simple example from mechanics.     

Statistical analysis of hierarchical stochastic models: Examples and approaches

This paper introduces and illustrates the concept of hierarchical or random parameter stochastic process models. These models arise when members of a population each generate a stochastic process governed by certain parameters and the values of the parameters may be viewed as single realizations of random variables. The paper treats the estimation of the individual parameter values and the parameters of the superpopulation distribution. Examples from system reliability, pharmacokinetic compartment models, and criminal careers are introduced; a reliability (Poisson process-exponential interval) process is examined in greater detail. An explicit, approximate, robust estimator of individual (log) failure rates is presented for the case of a long-tailed (Studentt) superpopulation. This estimator exhibits desirable limited shrinkage properties, refusing to borrow unjustified strength. Numerical properties of such estimators are described more fully elsewhere.     

A hybrid differential evolution algorithm to vehicle routing problem with fuzzy demands

In this paper, the vehicle routing problem with fuzzy demands (VRPFD) is considered, and a fuzzy chance constrained program model is designed, based on fuzzy credibility theory. Then stochastic simulation and differential evolution algorithm are integrated to design a hybrid intelligent algorithm to solve the fuzzy chance constrained program model. Moreover, the influence of the dispatcher preference index on the final objective of the problem is discussed using stochastic simulation, and the best value of the dispatcher preference index is obtained.     

A class of stochastic processes with a law generalizing a nondecomposable law on the real line

In this paper we define a class of stochastic processes where law can be considered as a natural generalization of a nondecomposable law. In particular case, we express the processes thus defined as semimartingales with a Brownian martingale part, and compute the likelihood for detecting a signal immersed in additive noise which looks like Brownian motion, but has different independence properties.     

Application of fuzzy decision making to the fuzzy finite element method

In this paper we present a new approach to handle uncertainty in the Finite Element Method. As this technique is widely used to tackle real-life design problems, it is also very prone to parameter-uncertainty. It is hard to make a good decision regarding design optimization if no claim can be made with respect to the outcome of the simulation. We propose an approach that combines several techniques in order to offer a total order on the possible design choices, taking the inherent fuzziness into account. Additionally we propose a more efficient ordering procedure to build a total order on fuzzy numbers.     

On the Geometric Densities of Random Closed Sets

Abstract

In many applications it is of great importance to handle evolution equations about random closed sets of different (even though integer) Hausdorff dimensions, including local information about initial conditions and growth parameters. Following a standard approach in geometric measure theory such sets may be described in terms of suitable measures. For a random closed set of lower dimension with respect to the environment space, the relevant measures induced by its realizations are singular with respect to the Lebesgue measure, and so their usual Radon–Nikodym derivatives are zero almost everywhere. In this paper we suggest to cope with these difficulties by introducing random generalized densities (distributions) á la Dirac–Schwarz, for both the deterministic case and the stochastic case. In this last one we analyze mean generalized densities, and relate them to densities of the expected values of the relevant measures. Many models of interest in material science and in biomedicine are based on time dependent random closed sets, as the ones describing the evolution of (possibly space and time inhomogeneous) growth processes; in such a situation, the Delta formalism provides a natural framework for deriving evolution equations for mean densities at all (integer) Hausdorff dimensions, in terms of the local relevant kinetic parameters of birth and growth. In this context connections with the concepts of hazard function, and spherical contact distribution function are offered.     

Some result on stochastic partial differential equations by the stochastic characteristics method

We study a class of second order (in the drift term) stochastic partial differential equations by the stochastic characteristics method, as developped by Kunita for the first order stochastic partial differential equations. With this method the original problem is transformed in a family of deterministic parabolic problems.     

Functional convergence of stochastic integrals with application to statistical inference

Assuming that {(U_n,V_n)} is a sequence of càdlàg processes converging in distribution to (U,V) in the Skorohod topology, conditions are given under which {?f_n(β,u,v)dU_ndV_n} converges weakly to ?f(β,x,y)dUdV in the space C(R), where f_n(β,u,v) is a sequence of “smooth” functions converging to f(β,u,v). Integrals of this form arise as the objective function for inference about a parameter β in a stochastic model. Convergence of these integrals play a key role in describing the asymptotics of the estimator of β which optimizes the objective function. We illustrate this with a moving average process.     

Decision making with imprecise probabilities and utilities by means of statistical preference and stochastic dominance

A problem of decision making under uncertainty in which the choice must be made between two sets of alternatives instead of two single ones is considered. A number of choice rules are proposed and their main properties are investigated, focusing particularly on the generalizations of stochastic dominance and statistical preference. The particular cases where imprecision is present in the utilities or in the beliefs associated to two alternatives are considered.     

Exponential synchronization of stochastic fuzzy cellular neural networks with time delay in the leakage term and reaction-diffusion

Separate studies have been published on the stability of fuzzy cellular neural networks with time delay in the leakage term and synchronization issue of coupled chaotic neural networks with stochastic perturbation and reaction-diffusion effects. However, there have not been studies that integrate the two fields. Motivated by the achievements from both fields, this paper considers the exponential synchronization problem of coupled chaotic fuzzy cellular neural networks with stochastic noise perturbation, time delay in the leakage term and reaction-diffusion effects using linear feedback control. Lyapunov stability theory combining with stochastic analysis approaches are employed to derive sufficient criteria ensuring the coupled chaotic fuzzy neural networks to be exponentially synchronized. This paper also presents an illustrative example and uses simulated results of this example to show the feasibility and effectiveness of the proposed scheme.     

Application of stochastic artsteins theorem to feedback stablization

The stochastic Artstein's theorem is applied to derive sufficient conditions for dynamic asymptotic stabilization in probability by means of a feedback integrator for a class of nonlinear stochastic differential systems. A stabilizing feedback law is deduced from a control Lyapunov function     

Existence of almost periodic solutions to some stochastic differential equations

The article introduces and studies the concept of p-mean almost periodicity for stochastic processes. Our abstract results are, subsequently, applied to studying the existence of square-mean almost periodic solutions to some semilinear stochastic equations.     

Probability Distribution of the Median Taken on Partial Sums of a Simple Random Walk

This article gives formulas for the probability distribution of the median taken on partial sums of a simple random walk. We also present an example in economics, where the median is interpreted as the price of a security in an informationally inefficient market.