Showing papers in "Stochastic Analysis and Applications in 1996"
TL;DR: In this paper, the Geral theory on the almost sure exponential stability and instability of the stochastically perturbed neural network is first established, and the theory is then applied to investigate stochastic stabilization and destabilization of the network.
Abstract: In this paper we shall discuss stochastic effects to the stability property of a neural network Suppose the stochastically perturbed neural network is described by an Ito equation [ILM002]The Geral theory on the almost sure exponential stability and instability of the stochastically perturbed neural network is first established. The theory is then applied to investigate the stochastic stabilization and destabilization of the neural network. Several interesting examples are also given for illustration
190 citations
TL;DR: In this article, the existence and uniqueness of an adapted solution for a linear, infinite dimensional, stochastic differential equation with final time condition was proved under very general assumptions on the coefficients.
Abstract: The existence and uniqueness of an adapted solution is proved, under very general assumptions on the coefficients, for a linear, infinite dimensional, stochastic differential equation with final time condition. Moreover the attention is focused on the regularity “in space” of this Solutions
64 citations
TL;DR: In this article, simple expressions for the raw moments and cumulants, of arbitrary order, of a quadratic form Q(x) in normal variates on R pand of the expectation of a product Q k of an arbitrary number of Quadratic forms in normally distributed variables
Abstract: Let x be normally distributed N p (μ∑) where ∑is positive definite and define the quadratic form and product of quadratic forms. We present explicite simple expressions for the raw moments and cumulants, of arbitrary order, of a quadratic form Q(x) in normal variates on R pand of the expectation of a product Q kof an arbitrary number of quadratic forms in normally distributed variables
38 citations
TL;DR: In this article, the maximum principle for optimal control of stochastic systems of functional type is obtained and the adjoint equation which seems to be new in form and proved the existence and uniqueness of the solution of that equation.
Abstract: In this paper, the maximum principle for optimal control of stochastic systems of functional type is obtained. We also derived the adjoint equation which seems to be new in form and proved the existence and uniqueness of the solution of that equation.
35 citations
TL;DR: In this article, the mean square stability of specific linear stochastic systems is studied and several results concerning asymptotical mean square stabilisation of equilibria are presented and proven.
Abstract: Several results concerning asymptotical mean square stability of equilibria of specific linear stochastic systems are presented and proven. These discrete time systems can be interpreted as numerical solution of stochastic differential equations driven by Wiener noise. Effects of the presented mean square calculus are shown by the Kubo oscillator perturbed by white noise and a simplified system of noisy Brusselator equations
35 citations
TL;DR: In this article, a large system of particles is studied and its time evolution is determined as the superposition of two components: the independent motion of each particle and the random interaction mechanism between pairs of particles.
Abstract: A large system of particles is studied. Its time evolution is determined as the superposition of two components. The first component is the independent motion of each particle. The second component is the random interaction mechanism between pairs of particles. The intensity of the interaction depends on the state of the system and is assumed to be bounded Convergence of the empirical measures is proved as the number of particles tends to infinity. The limiting deterministic measure-valued function is characterized as the unique solution of a nonlinear equation of the Boltzmann type
21 citations
TL;DR: In this paper, the solution of the 1-dimensional heat equation with white noise potential and its asymptotic behavior was constructed. But the authors did not consider the white noise in the solution.
Abstract: (1996). Construction of the solution of 1-dimensional heat equation with white noise potential and its asymptotic behaviour. Stochastic Analysis and Applications: Vol. 14, No. 4, pp. 487-506.
19 citations
TL;DR: Fuzziness is discussed in the context of fuzzy random variable and a corresponding view of fuzzy martingale and some their properties are given.
Abstract: Fuzziness is discussed in the context of fuzzy random variable and a corresponding view of fuzzy martingale and some their properties are given. Fuzzy random variables are introduced as random variables whose values are not reals but fuzzy sets. In this paper we study fuzzy valued random processes in discrete time and with values in a set of fuzzy sets of separable Banach space.
15 citations
TL;DR: In this article, the existence and uniqueness of solutions of the Cauchy problem and the related elliptic problem are proved in such a Gauss-Sobolev space setting.
Abstract: We consider the Kolmogorov equation in infinite dimensions associated with a nonlinear stochastic evolution equation in some Hilbert space. By introducing a proper class of Gauss-Sobolev spaces, a L2 -theory of the Kolmogorov equation is developed. Under some suitable conditions, the existence and uniqueness of solutions of the Cauchy problem and the related elliptic problem are proved in such a Gauss-Sobolev space setting
10 citations
TL;DR: In this article, it was shown that for all adaptive rules π,,, for all closed sets for all open sets where I(x) and J(x), are rate functions independent of π, then, if π is finite in a neighborhood of θ = 0, then
Abstract: Let be r sequences of i.i.d. random variables and the sample mean of an n-size sample, given an adaptive allocation rule . We show that if is finite in a neighborhood of θ= 0, then, for all adaptive rules π, , for all closed sets for all open sets where I(x) and J(x) are rate functions independent of π
9 citations
TL;DR: In this article, an extremal problem for the functional where X is a Banach spase, μ is a smooth measure on X, Ais a.m.map from a functional space B 1(X) to a functional spaces B 2(X), and extend the main results of classical calculus of variations to the case under consideration is considered.
Abstract: We consider an extremal problem for the functional where X is a Banach spase, μ is a smooth measure on X, Ais a.map from a functional space B 1(X) to a functional space B 2(X), and extend the main results of classical calculus of variations to the case under consideration. The infinite-dimensional analogs of the Euler-Lagrange equation, the Noether theorem, the canonical Hamilton system are obtained. The illustrations of these results for (where z(x) is a vector field) are given. The example related to the stochastic optimal control theory is considered
TL;DR: In this article, a convergence in distribution theorem is proved for the solution of the stochastic Navier-Stokes equation with multiplicative noise in dimension 2 or 3 when the noise is a mixing process.
Abstract: A convergence in distribution theorem is proved for the solution of the stochastic Navier-Stokes equation with multiplicative noise in dimension 2 or 3 when the noise is a mixing process. This result generalizes previous diffusionapproximation theorems to a non-linear case
TL;DR: In this article, the moments of central and noncentral Wishart distributions are obtained by differentiating their characteristic functions applying matrix derivative techniques, using a special operator which takes into account the symmetry of the matrices.
Abstract: Moments of central and noncentral Wishart distributions are obtained by differentiating their characteristic functions applying matrix derivative techniques, using a special operator which takes into account the symmetry of the matrices. As a special case, higher moments of the multivariate normal distribution are obtained, arranged automatically in a square matrix form
TL;DR: In this paper, the expected number of up-crossings with slope greater than udown-crossing with slope less than −1 of a Gaussian process ξ(t) where uis any positive constant was provided.
Abstract: In this paper we provide the expected number of zero up-crossings with slope greater than udown-crossing with slope less than —uof a Gaussian process ξ(t) Where uis any positive constant. Promoted by graphical interpretation, we define hese crossings as u—sharp. Then the expected number of such crossings of a random lgebraic polynomial of the form with normally distributed coefficients follows from this result. It is Shown that for any bounded uthe expected number of u-sharp crossings is asymptotically equal to 0-sharp crossings while for u→ ∞ as n→ ∞ such that (u 2/3/n)→0 the expected number in he interval (-1,1) asymptotically remains as (1/π) log nand, outside this interval, asymtotically reduces to
TL;DR: In this paper, a random approximation theorem for multivalued nonexpansive random operator defined on an unbounded subset of a Hilbert space is proved, and a random fixed point theorem is derived.
Abstract: A random approximation theorem for multivalued nonexpansive random operator defined on an unbounded subset of a Hilbert spaces is proved. As an application of our theorem, a random fixed point theorem is derived
TL;DR: In this article, a finite state birth and death process with absorbing barriers is carried out and closed form expression for state probabilities and density functions of time of absorption and first passage time for each case is obtained along with the moments.
Abstract: Transient analysis of finite state birth and death process with absorbing barriers is carries out. Closed form expression for state probabilities and density functions of time of absorption and first passage time for each case is obtained along with the moments. The results are expressed in terms of eigenvalues of symmetric triadiagonal matrix which can be easily computed
TL;DR: In this paper, the authors studied the regularity of the heat semigroup generated by the OperatorH=L+V where Vis a potential bounded from above, and Lis the generator of the Ornstein-Uhlenbeck process ξ which is the solution of the equation [ILM0001] where B(t)is a cylindrical Brownian motion on a Hilbert space HA -2is an unbounded operator.
Abstract: First, we study the regularity of the heat semigroup generated by the OperatorH=L+Vwhere Vis a potential bounded from above, and Lis the generator of the Ornstein-Uhlenbeck process ξwhich is the solution of the equation [ILM0001}where B(t)is a cylindrical Brownian motion on a Hilbert space HA -2is an unbounded operator. In the final section, we study the regularity of the transition semigroup associated with the stochastic evolution equation where Wis a vector field on H.
TL;DR: In this article, it was shown that singular diffeomorphisms asymptotically nullify the arc-length of oriented rectifiable arcs and recover one of their earlier results on the flatness of singular diffusions.
Abstract: LetX t (x) be a singular Process describing a flow of diffeomorphisms ,and kbe a compact surface in with positive finite Hausdorff measure. We present conditions under which the area of Xt:X (k)goes to zero almost surely and in moments as t→∞. It is shown, in particular, that the flowXtX (·) asymptotically nullifies the arc-length of oriented rectifiable arcs . As a consequence, we recover one of our earlier results on the asymptotic flatness of singular diffusions. Though we work with degenerate diffusions, our method applies to the nondegenerate case as well
TL;DR: The metrical properties of the unique stationary law for the one-step predictor of a finite state Markov Chain from noisy observations and a lower bound for the dimension of S is presented, showing that the stationary law is singular with respect to the Lebesgue measure.
Abstract: We analyze metrical properties of the unique stationary law for the one-step predictor of a finite state Markov Chain from noisy observations. In Piccioni (1990), the topological aspect of this problem was analyzed. Our work is a natural follow-up of this paper. We will be concerned with the case where the stationary law has support in a totally disconnected and perfect set. In this case the predictor keeps an infinite memory of the past observations. The closure of the support of this stationary law is called the attractor S(Elton and Piccioni (1992)). We present a lower bound for the dimension of S. This lower bound will be also an upper bound for the exponent scale of the law. As a consequence of our results, we partially answer a question raised by Piccioni (1990), in a case (b=c, see notation in Section 4) where the closure of the invariant measure's support is an interval, showing that the stationary law is singular with respect to the Lebesgue measure
TL;DR: In this article, it was shown that if the equilibrium solution of a nonlinear control stochastic system is locally stable in probability by means of a continuous state feedback law, then the resulting system obtained by adding an integrator is also locally asymptotically stable.
Abstract: The aim of this work is to prove that if the equilibrium solution of a nonlinear control stochastic system is locally asymptotically stable in probability by means of a continuous state feedback law, then the resulting stochastic system obtained by adding an integrator is also locally asymptotically stable in probability by means of a smooth, except possibly at the equilibrium solution, state feedback law. This result extends to the stabilization of stochastic systems a result proved by Tsinias [9] for deterministic systems. In our proof, we make use of the stochastic version of Artstein's theorem established in [4]
TL;DR: The stochastic extension of an algorithm of Bonnans to solve a control of diffusion problem and results concerning the convergence of this algorithm are obtained.
Abstract: In this paper, we introduce the stochastic extension of an algorithm of Bonnans to solve a control of diffusion problem. We also obtain results concerning the convergence of this algorithm
TL;DR: In this paper, the integral solution of the following nonlinear stochastic evolution equation is defined, and it is shown that there exists a unique integral solution for this equation under some condition for A and Ф (t).
Abstract: In this paper, we define the integral solution of the following nonlinear stochastic evolution equation .and we prove that there exists a unique integral solution of this equation some condition for A and Ф (t).
TL;DR: In this paper, local or global solutions to first order stochastic functional partial differential equations of Ito type are investigated based on the characteristics and the successive approximations methods.
Abstract: Local or global solutions to first order stochastic functional partial differential equations of Ito type are investigated. The proofs are based on the characteristics and the successive approximations methods.The formulation includes retarded arguments and hereditary Volterra terms
TL;DR: In this article, the authors consider a one-dimensional diffusion X on a finite or infinite interval satisfying a stochastic differential equation with W(t) t 0 standard Brownian motion.
Abstract: We consider a one-dimensional diffusion X on a finite or infinite interval]l,r[ satisfying a stochastic differential equation with W(t) t 0 standard Brownian motion. For fixed drift b we investigate noiseinduced transitions of the boundary behavior i.e. the dependence of the behavior of X near r when σ varies. Particularly we are interested in the question whether increasing σ (for all x) may cause or prevent explosion in finite time. A negative result is that if σ is bounded and bounded away from zero then multiplying σ by a factor greater than one can never prevent explosions (Theorem 2.1). Various examples show that under different assumptions increasing the noise can have a drastic stabilizing or destabilizing effect (depending on b and σ). In the last section we study the relation between the stochastic and deterministic (σ ≡ 0) case
TL;DR: In this paper, the authors considered the pathwise average cost per unit problem for a queueing network in heavy traffic and showed that the scaled controlled reflected system converges weakly to a controlled limit reflected diffusion.
Abstract: The pathwise average cost per unit problem for a queueing network in heavy traffic is considered. Various input and service interruptions are the controls. We show that the scaled controlled reflected system converges weakly to a controlled limit reflected diffusion. It is also proved that the optimal policies for the limit problem when adapted to the physical system are nearly optimal. The martingale problem methods are utilized in the analysis. An approach based on functional occupation measures is used
TL;DR: In this paper, the method of quasilinearization was extended to stochastic initial value problems, and the method was used to solve the problem of initial value maximization.
Abstract: In this paper, we extend the method of quasilinearization to stochastic initial value problems
TL;DR: In this article, the authors identify a class of non-stationary vector ARMA processes with one unit characteristic root that can be used to model multiple time series that stablize together.
Abstract: This paper identifies a class of non-stationary vector ARMA processes with one unit characteristic root that can be used to model multiple time series that stablize together. Analysis of ARMA processes with unit roots is generally more difficult because of their probabilistic roperties. In particular, general theory of law of large numbers and central limit theory do not apply to tandardized sums of the realizations of these processes. However, this paper establishes a weak law of large numbers for a useful class of such processes.
TL;DR: In this paper, the theory of stochastic differential inequalities is systematically developed and sufficient conditions for stability in probability, with probability one and in the mean of the Ito-type, are given, using the method of cone-valued Lyapunov functions.
Abstract: Stochastic differential equations of Ito-type are considered and the theory of stochastic differential inequalities is systematically developed. Sufficient conditions for stability in probability, with probability one and in the mean of the Ito-type stochastic differential equations are given, using the method of cone-valued Lyapunov functions. Necessary and sufficient conditions for the construction of stochastic cone-valued Lyapunov functions are obtained for the cases where the Ito-type stochastic differential equations have uniform asymptotic stability in probability and uniform asymptotic stability in the mean. The results obtained are applied to the study of connective stability of multispecies community in a stochastic environment
TL;DR: In this article, a correction to the proof of the main result in [3] was presented, and the following theorem was established: the main theorem is correct in this paper.
Abstract: In this short note, we present a correction to the proof of the main result in [3] In this short note, we establish the following theorem