Pathwise uniqueness for solutions of systems of stochastic differential equations

A general theorem which obtains pathwise uniqueness for solutions of systems of Ito stochastic differential equations is given. It is shown that this theorem contains as special cases basic criteria which generalize Ito's result in which the coefficients satisfy Lipschitz conditions in the second variable. Also some new results which assume t-dependent modulus of continuity conditions on the coefficients are given as corollaries. The main result is established by means of Lyapunov type functions and comparison principle techniques.     

Pathwise random periodic solutions of stochastic differential equations

In this paper, we study the existence of random periodic solutions for semilinear stochastic differential equations. We identify these as the solutions of coupled forward-backward infinite horizon stochastic integral equations in general cases. We then use the argument of the relative compactness of Wiener-Sobolev spaces in C⁰([0,T],L²(Ω)) and generalized Schauder?s fixed point theorem to prove the existence of a solution of the coupled stochastic forward-backward infinite horizon integral equations. The condition on F is then further weakened by applying the coupling method of forward and backward Gronwall inequalities. The results are also valid for stationary solutions as a special case when the period τ can be an arbitrary number.     

Heavy traffic approximation for the stationary distribution of stochastic fluid networks

It has recently been shown that in the heavy traffic limit, the stationary distribution of the scaled queue length process of a Generalized Jackson Network converges to the stationary distribution of its corresponding Reflected Brownian Motion limit. In this paper, we show that this “interchange of limits” is valid for Stochastic Fluid Networks with Lévy inputs. Furthermore, under additional assumptions, we extend the result to show that the interchange is valid for moments of the stationary distribution and for state-dependent routing. The results are obtained using monotonicity and sample-path arguments.     

Pathwise convergence rates for numerical solutions of Markovian switching stochastic differential equations

This work develops numerical approximation algorithms for solutions of stochastic differential equations with Markovian switching. The existing numerical algorithms all use a discrete-time Markov chain for the approximation of the continuous-time Markov chain. In contrast, we generate the continuous-time Markov chain directly, and then use its skeleton process in the approximation algorithm. Focusing on weak approximation, we take a re-embedding approach, and define the approximation and the solution to the switching stochastic differential equation on the same space. In our approximation, we use a sequence of independent and identically distributed (i.i.d.) random variables in lieu of the common practice of using Brownian increments. By virtue of the strong invariance principle, we ascertain rates of convergence in the pathwise sense for the weak approximation scheme.     

The exponential integrator scheme for stochastic partial differential equations: Pathwise error bounds

We present an error analysis for the pathwise approximation of a general semilinear stochastic evolution equation in d dimensions. We discretise in space by a Galerkin method and in time by using a stochastic exponential integrator. We show that for spatially regular (smooth) noise the number of nodes needed for the noise can be reduced and that the rate of convergence degrades as the regularity of the noise reduces (and the noise becomes rougher).     

一类(QL)型随机微分方程解的轨道唯一性判别及其应用

考虑了一类拟左连续(QL)型随机微分方程(S.D.E.)解的轨道唯一性,应用随机分析方法获得了唯一性成立的一般判别定理,并在方程系数满足局部(或非)Lipschitz条件下给出了一些应用实例.     

Stability results for stochastic delayed recurrent neural networks with discrete and distributed delays

We present new conditions for asymptotic stability and exponential stability of a class of stochastic recurrent neural networks with discrete and distributed time varying delays. Our approach is based on the method using fixed point theory, which do not resort to any Liapunov function or Liapunov functional. Our results neither require the boundedness, monotonicity and differentiability of the activation functions nor differentiability of the time varying delays. In particular, a class of neural networks without stochastic perturbations is also considered. Examples are given to illustrate our main results.     

Circulant preconditioners for stochastic automata networks

Summary. Stochastic Automata Networks (SANs) are widely used in modeling communication systems, manufacturing systems and computer systems. The SAN approach gives a more compact and efficient representation of the network when compared to the stochastic Petri nets approach. To find the steady state distribution of SANs, it requires solutions of linear systems involving the generator matrices of the SANs. Very often, direct methods such as the LU decomposition are inefficient because of the huge size of the generator matrices. An efficient algorithm should make use of the structure of the matrices. Iterative methods such as the conjugate gradient methods are possible choices. However, their convergence rates are slow in general and preconditioning is required. We note that the MILU and MINV based preconditioners are not appropriate because of their expensive construction cost. In this paper, we consider preconditioners obtained by circulant approximations of SANs. They have low construction cost and can be inverted efficiently. We prove that if only one of the automata is large in size compared to the others, then the preconditioned system of the normal equations will converge very fast. Numerical results for three different SANs solved by CGS are given to illustrate the fast convergence of our method. Received March 17, 1998 / Revised version received August 16, 1999 / Published online July 12, 2000     

Diffusion approximation for signaling stochastic networks

This paper introduces an unified approach to diffusion approximations of signaling networks. This is accomplished by the characterization of a broad class of networks that can be described by a set of quantities which suffer exchanges stochastically in time. We call this class stochastic Petri nets with probabilistic transitions, since it is described as a stochastic Petri net but allows a finite set of random outcomes for each transition. This extension permits effects on the network which are commonly interpreted as “routing” in queueing systems. The class is general enough to include, for instance, G-networks with negative customers and triggers as a particular case. With this class at hand, we derive a heavy traffic approximation, where the processes that drive the transitions are given by state-dependent Poisson-type processes and where the probabilities of the random outcomes are also state-dependent. The objective of this approach is to have a diffusion approximation which can be readily applied in several practical problems. We illustrate the use of the results with some numerical experiments.     

Stability results for stochastic programming problems

The paper deals with a statistical approach to stability analysis in nonlinear stochastic programming. Firstly the distribution function of the underlying random variable is estimated by the empirical distribution function, and secondly the problem of estimated parameters is considered. In both the cases the probability that the solution set of the approximate problem, is not contained in an l-neighbourhood of the solution set to the original problem is estimated, and under differentiability properties an asymptotic expansion for the density of the (unique) solution to the approximate problem is derived.     

Pathwise rate- stability for input-output processes

An input-output processZ = {Z(t), t 0} is said to be-rate stable ifZ(t) = o((t)) for some non-negative function(t). We prove that the processZ is -rate stable under weak conditions that include the assumption that input satisfies a linear burstiness condition and Z is asymptotically average stable. In many cases of interest, the conditions for-rate-stability can be verified from input data. For example, using input information, we establish-rate stability of the workload for multiserver queues, an ATM multiplexer, and-rate stability of queue-length processes for infinite server queues.     

Existence results for optimal stochastic controls

The paper treats the problem of existence of optimal controls for a large class of delay-differential Itô equations, where the control is a nonanticipative measurable function of the trajectory (the case of complete information). The technique, which seems simpler than past approaches to the problem, requires the use of results on weak convergence of measures, and gives fairly general results. Control can be either over a fixed-time interval, or it can terminate when a target set is reached, and there can be additional (almost everywhere continuous) side constraints.     

Further results on mean square exponential stability of uncertain stochastic delayed neural networks

In this paper, the mean square exponential stability problem is deal with for a class of uncertain stochastic neural networks with time-varying delays. By introducing a new Lyapunov–Krasovskii function, improved delay-dependent stability criteria are established in term of linear matrix inequalities (LMIs). Finally, two numerical examples are given to show that our results are less conservative and more efficiency than the existing stability criteria.     

Multi-stage stochastic fluid models for congestion control

We consider multi-stage stochastic fluid models (SFMs), driven by applications in telecommunications and manufacturing in which control of the behavior of the system during congestion may be required. In a two-stage SFM, the process starts from Stage 1 in level 0, and moves to Stage 2 when reaching threshold _b2$b_2$ from below. Stage 1 starts again when reaching threshold _b1<_b2$b_1<b_2$ from above. While in a particular stage, the process evolves according to a traditional SFM with a unique set of phases, generator and fluid rates. We first consider a two-stage SFM with general, real fluid change rates. Next, we analyze a two-stage SFM with an upper boundary B>_b2$B>b_2$. Finally, we discuss a generalization to multi-stage SFMs. We use matrix-analytic methods and derive efficient methodology for the analysis of this class of models.     

Robust stability for discrete-time stochastic genetic regulatory networks

Attention in this paper is focused on the study of the problem of asymptotic stability for a class of discrete-time stochastic genetic regulatory networks with time-varying but norm-bounded parameter uncertainties. By the Lyapunov–Krasovskii functional approach, delay-dependent stability criteria are derived in terms of linear matrix inequalities. Simulation examples are provided to show the effectiveness of the proposed results.     

Pathwise global attractors for stationary random dynamical systems

Summary The asymptotic behaviour of random dynamical systems in Polish spaces is considered. Under the assumption of existence of a random compact absorbing set, assumption supposed to hold path by path, a candidate pathwise attractorA() is defined. The goal of the paper is to show that, in the case of stationary dynamical systems,A() attracts bounded sets, is measurable with respect to the -algebra of invariant sets, and is independent of when the system is ergodic. An application to a general class of Navier-Stokes type equations perturbed by a multiplicative ergodic real noise is discussed in detail.     

Convergence results for continuous-time adaptive stochastic filtering algorithms

The adaptive stochastic filtering problem for Gaussian processes is considered. The self-tuning synthesis procedure is used to derive two algorithms for this problem. Almost sure convergence for the parameter estimate and the filtering error will be established. The convergence analysis is based on an almost-supermartingale convergence lemma that allows a stochastic Lyapunov-like approach.     

Multiscale diffusion approximations for stochastic networks in heavy traffic

Stochastic networks with time varying arrival and service rates and routing structure are studied. Time variations are governed by, in addition to the state of the system, two independent finite state Markov processes X and Y. The transition times of X are significantly smaller than typical inter-arrival and processing times whereas the reverse is true for the Markov process Y. By introducing a suitable scaling parameter one can model such a system using a hierarchy of time scales. Diffusion approximations for such multiscale systems are established under a suitable heavy traffic condition. In particular, it is shown that, under certain conditions, properly normalized buffer content processes converge weakly to a reflected diffusion. The drift and diffusion coefficients of this limit model are functions of the state process, the invariant distribution of X, and a finite state Markov process which is independent of the driving Brownian motion.