Showing papers in "Stochastic Analysis and Applications in 1984"

A stopped Doob inequality for stochastic convolution integrals and stochastic evolution equations

Abstract: A stopped Doob inequality is proved for stochastic convolution integrals in Hilbert space, where M is a square integrable Hilbert space valued cadlag martingale, ⊘ an operator valued predictable function and U(t, s) a contraction-type evolution operator. This allows to obtain the mild solution for evolution equations (with memory) where A(t) is the quasigenerator of U(t,s), V(t) a bounded variation process, and Z(t) a semimartingale, under the same weak assumptions on B and D as for stochastic differential equations with bounded coefficients, i .e ., (A( t) = 0) . Moreover, a weak convergence criterion for is derived.

An estimate of Burkholder type for stochastic processes defined by the stochastic integral

Abstract: We develop a method to prove an estimate of Burkholder type for a class of processes defined by a stochastic convolution with a semigroup for p ≥ 2; this result, for p = 2, has been obtained also by Kotelenez [6] using a martingale inequality due to him[5]

Asymptotic analysis of P.D.E.s with wide–band noise disturbances, and expansion of the moments

Abstract: We present an asymptotic analysis -in the “ white-noise limit”- of a linear parabolic partial differential equation, whose coefficients are perturbed by a wide-band noise. After having studied some ergodic properties of a class of diffusion processes, we prove the convergence in law towards the solution of an Ito stochastic P.D.E. We then establish an expansion in powers of Δ ( 1/Δ being a measure of the bandwith of the driving noise) of the first moment of the solution

An approach to Ito linear equations in Hilbert spaces by approximation of white noise with coloured noise

Abstract: We consider the stochastic problem , in a Hilbert space H, where f,X are prescribed data, Wt is a real Brownian motion, and A(t), B generate an analytic semi-group and a strongly continuous group respectively. The domains D(A (t)) may vary with t and we only require D(A(t))CD(B) for each t. A unique generalized solution is constructed as the pathwise uniform limit of solutions of suitable approximating deterministic problems, which are obtained by approaching the white noise dWt with a sequence of regular coloured noises

Stochastic stability of coupled linear systems: a survey of methods and results

Abstract: The analysis of dynamic systems subject to stochastic parametric excitation is important in a variety of branches of engineering and physics. For example, models of this type frequently occur in the analysis of linear continuous systems using modal decomposition. The random coupling or parametric excitation can, for example, model the influence of externally applied loads on the system parameters. In this paper we investigate the almost–sure stability properties of the sample trajectories of linear stochastic systems with parametric excitation.

Stochastic convergence of randomly weighted sums of random elements

Abstract: Let {Xn} be random elements in a separable Banach space and let {ank} be an array of random variables. Convergence in probability and almost surely is obtained for the weighted sum under varying distributional and moment conditions on the random weights and on the random elements and geometric conditions on Banach spaces. In general, these results include the results for constant weights and real-valued random variables and are motivated in part by estimation problems and consistency considerations. Moreover, similar results are obtained for the space D[0, 1] under varying hypotheses of boundedness conditions on the moments and conditions on the mean oscillation of the random elements {Xn} on subintervals of a partition of [0,1] and represent significant improvements over existing laws o f large numbers and convergence results for weighted sums of random elements in D[0,1].

Goal uncertainity in a generalized information system: convergence properties of the estimated expected utilities ∗

Abstract: The concept of goal uncertainty in a generalized information system is modeled by treating utility functions as interval valued rather than real valued. Furthermore, since the system is dynamic, the intervals shrink as information accumulates. The lower (upper) endpoints of these intervals are shown to be sub(super) martingales. Decision rules are formulated to select the appropriate course of action for each cycle through the decision-information feedback loop which is the heart of this information system. Finally, the theory of multifunctions is used to obtain convergence theorems for the estimated expected utilities generated by this system

A note on expectation of a random quadratic form

Abstract: Let be a random vector with . Useful expressions are given for expectation of quadratic forms in in terms of ∧ explicitly

Existence of solutions and stability of a class of parabolic systems perturbed by generalized white noise on the boundary

Abstract: This paper is concerned with a class of stochastic boundary value problems and their stability questions. The system, we consider, is governed by a parabolic partial differential equation perturbed by generalized white noise on the boundary. Existence of weak solutions and their regularity properties are established. It is also shown that the solution of the autonomous system generates a Feller process in a Hilbert space, in case the spatial operator is time invariant. The questions of Lyapunov type stability of this class of systems are also examined. The system is shown to be almost surely globally asymptotically stable with respect to a ball centered at the origin. Further, it is shown that there exists a measure, supported on the attractor, which is invariant with respect to the adjoint Feller semigroup. An explicit expression for the generator of the semigroup is also given

An improved lower bound for the expected number of real zeros of a random algebraic polynomials

Abstract: In this note we estimate the lower bound of the average number of real zeros of a random algebraic polynomials when the random coefficients are standard normal random variables

The existence and uniqueness of solutions to stochastic differnetial–difference equations

Abstract: The concept of an integral contractor is utilized to obtain very general conditions for the existence and uniqueness of solutions to a stochastic differential-difference equation of the form for t ≥ 0, In this equation w(t) is a Weiner process and x(t) denotes a stochastic process.

The brownian theory of light

Abstract: Probabilistic origin of the laws of Mechanics and Relativity: In a diffusion with exclusion principle, the brownian bridges propagate pressure wavelets which interfere and make wave packets of which the group velocity Vg tends to equal the expansion velocity v of the Huygens sphere envelope of these wavelets. This convergence Vg > v due to the strong large numbers law involves the Galilean and Lorentz groups, the limit Vg =v is reached on monochromatic waves and stationary states. The fundamental structure involved, open or closed, is to be found also in the central nervous system of Man, emerging through Evolution.

On Stochastic Optimality of Policies in First Passage Problems.

Abstract: In many stochastic scheduling and optimal maintenance problems that have been consisdered in the literature, the optimization criterion employed has often been equivalent to minimizing the expected first passage times to a set of “desirable” states. A typical method that has been used in establishing the optimality of a certain policy is the method of successive approximations. As an intermediate, reuslt, this techinique often horizon versions of the problem. In this paper we point out that under mild assumptions stochastically, i.e. they are optimal in expectation for all members of a sequence of appropriately defined finite horizon problems. Furthermore, this characterization can reduce a “uniformizable” continuous time problem into a sequence of approximately defined discrete time problems. In the final sections we use this characterization to establish the stochastic optimality of pertinent policies for an optimal repair allocation problem and for a scheduling problem

Hedge portfolios and the black-scholes equations

Abstract: In 1973, Black and Scholes showed that a portfolio made up of shares of an asset A, whose price varies as a geometric Brownian motion, and shares of an asset B, whose price per share is functionally dependent on the price per share of A could be manipulated to be riskless, and designed to achieve any given rate of return on investment In this paper, we extend the results of Black and Scholes, by showing that their results remain valid when the geometrical Brownian motion assumption on the price of A is substantially relaxed, and when the number of assets in the portfolio is larger than two.

Non–linear filtering of diffusion processes with discontinuous observations

Abstract: Let xt be a diffusion process satisfying a stochastic differential equation of the form where [Wtilde] is an n-dimensional Brownian motion. Let the observed process yt be related to xt by ,where W is one dimensional Brownian motion Independent of [Wtilde]. The measure γ is a random counting measure independent of [Wtilde] and W. The problem is to find the conditional probability of the process xt given the observed path yt 0. The results of absolute continuity of measures are used to derive a stochastic differential equation for the required conditional probability .Our results are easily extended to the case where the process xt is governed by a stochastic differential equation also containing jump process, as indicated in Remark 1. Our results also cover the filter equation given by Gertner [4], Di Masi [5] and Pardoux [6] . Further, Zakaitype equation (linear stochastic partial differential equation) corresponding to the systems considered by Shiryayev [3] and Snyder [9], which are of Kushner type (non...

Stochastic analysis of a n-unit repairable system with slow switch

Abstract: This paper deals with the costn–benefit analysis of a cold standby system composed of n identical repairable units, subject to slow switch. Two models of system functioning are studied in this paper. In model 1, the repair time of a unit is assumed to follow exponential distribution and the other time distributions as arbitrary, while in model 2, the repair time of a unit is assumed to be arbitrarily distributed and the other time distributions follow exponential law. For both the models, the system characteristics, namely (i) the expected upn–time of the system during the period (O,t] (ii) the expected busyn–period of the repair facility during the period (0,t] and (iii) the expected time the units spend in the switchover/installation state during the period (O,t] are studied by identifying the system a t suitable regeneration epochs. The cost-benefit analysis is carried out using these characteristics

Some weak and strong laws of large numbers for D[0,1]-valued random variables

Abstract: Pointwise Weak Law of Large Numbers and Weak Law of Large Numbers in the norm topology of D[0,l] are shown to be equivalent under uniform convex tightness and uniform integrability conditions for weighted sums of a sequence of random elements in D[0,1]. Uniform convex tightness and uniform integrability conditions are jointly characterized. Marcinkiewicz–Zygmund–Kolmogorov's and Brunk– Chung's Strong Laws of Large Numbers are derived in the setting of D[0,l]-space under uniform convex tightness and uniform integrability conditions. Equivalence of pointwise convergence, convergence in the Skorokhod topology and convergence in the norm topology f o r sequences in D[0,l] is studied

Stochastic analysis of A 1-server n-unit system subject to inspection and several failure modes

Abstract: The main intent of the paper is to investigate the stochastic behaviour of a single-server n-unit system subject to random inspection and several failure modes. The time between successive inspections is a random variable distributed exponentially. It is assumed that the life-time of a unit is arbitrarily distributed while all the other time distributicns involved are exponential. At any instant t, the system is characterized by the probabilities of its beinq in the ‘up’ or ‘down’ state . Integral equations are established for these probabilities by identifying the system at suitable regenerative epochs corresponding to different initial conditions. Various system parameters of significant importance, namely, 1. point-wise availability of the system at instant t, 2. steady-state availability of the system, 3. s-expected up-time of the system in [o, t], 4. s-expected inspection time of the server in [o, t], 5. s-expected repair time of type i (1 ≤ i ≤ r) in [o, t] and 6. s-expected net gain per unit time i...

On the nonexistence of solutions for random differential equations in banach spaces

Abstract: Random differential equations in Banach spaces without solutions are considered Some density results are established

Central limit theorems for weighted sums of exchangeable random elements in banach spaces (1)

Abstract: A generalization of a central limit theorem for martingale difference arrays due to D L McLeish is obtained for random elements in a separable Banach space E This result, with a technique of N C Weber, is used to obtain a weak convergence theorem for weighted sums of a row-wise exchangeable array (Xnk) of random elements in a Banach space E which is uniformly 2-smooth Corollaries include a central limit theorem for weighted sums of an exchangeable sequence (Xk), and several weak laws of large numbers for such weighted sums

Measures of Noncompactness and Fixed Points of Random Operations.

Abstract: : The purpose of this paper is to consider some measures of noncompactness and their application to fixed point theorems for random operators. In a subsequent paper we will consider applications of these results to random operator equations. This paper is not the first to consider measures of noncompactness in probabilistic analysis. We refer to the book of Constantin and Istratescu for results on measures of noncompactness and probabilistic metric spaces. Section 2 gives some basic definitions of measures of noncompactness, and lists their properties. Section 3 is devoted to applications of measures of noncompactness to fixed point theorems for random operators. Keywords include: Operators (Mathematics), Probability, Noncompactness.

Bound state problem for random potentials

Abstract: We give a Martin–like asymptotics of the expected number of bound states for Schrodinger operator where Tt is certain random transformation of V

The exit problem for certain dynamical systems with small poisson disturbances

Abstract: The mean first passage time from the inside to the outside of a finite interval for the solution of a differential equation perturbed by Poisson noise may have a discontinuity at one end of the interval. When the size ofthe perturbation is small, the magnitude of the discontinuity can be large and difficult to calculate. In this paper asymptotic approximations for the size of the discontinuity are derived in two cases of practical interest. The approximations show that the discontinuity grows exponentially.