Showing papers in "Stochastics and Stochastics Reports in 2000"

Some solvable stochastic control problems with delay

Abstract: We consider optimal harvesting of systems described by stochastic differential equations with delay. We focus on those situations where the value function of the harvesting problem depends on the initial path of the process in a simple way, namely through its value at 0 and through some weighted averages A verification theorem of variational inequality type is proved. This is applied to solve explicitly some classes of optimal harvesting delay problems

Asymptotic properties of neutral stochastic differential delay equations

Abstract: This paper discusses asymptotic properties, especially asymptotic stability of neutral stochastic differential delay equations. New techniques are developed to cope with the neutral delay case, and the results of this paper are more general than the author's earlier work within the delay equations

Finite-fuel singular control with discretionary stopping

Abstract: We discuss the finite-fuel, singular stochastic control problem of optimally tracking the standard Brownian motion started at , by an adapted process of bounded total variation , so as to minimize the total expected discounted cost over such processes and stopping times τ. Here , and are given real numbers. In its form this problem goes back to the seminal paper of Bene[sbreve], Shepp and Witsenhausen (1980). For fixed α>0 and δ>0 we characterize explicitly the optimal policy in the case λ>αδ (of the “act-or-stop” type, since the continuation cost is relatively large), and in the case with (of the “act, stop, or wait” type, since the relative continuation cost is relatively small). In the latter case, an associated free-boundary problem is solved exactly. The case , of “moderate” relative continuation cost, is suggested as an open question

Solving Wick-stochastic boundary value problems using a finite element method

Abstract: A class of linear elliptic Wick-stochastic boundary value problems is considere. The problems are formulated in a variational (or weak) form and existence and uniqueness of a solution to this variational formulation is proved under general assumptions on the data. Furthermore, a Galerkin type of finite element method is formulated and presented in an algorithmic form. As an illustration the algorithm is then applied to the Wick-stochastic pressure equation in one and two dimensions.

On function spaces related to fractional diffusions on d-sets

Abstract: We prove that all the Dirichlet forms associated with certain diffusions on a d-set are equivalent and that their common domain is an integral Lipschitz space. We also provide an analytic characterisation of the walk dimension dw of a d-set F and show that all fractional diffusions on F share dw as their walk dimension.

General approach to filtering with fractional brownian noises — application to linear systems

Abstract: At first a general approach is proposed to filtering in systems where the observation noise is a fractional Brownian motion. It is shown that the problem can be handled in terms of some appropriate semimartingale and analogs of the classical innovation process and fundamental filtering theorem are obtained. Then the problem of optimal filtering is completely solved for Gaussian linear systems with fractional Brownian noises. Closed form simple equations are derived both for the mean of the optimal filter and the variance of the filtering error. Finally the results are explicited in various specific cases

Optimal control of assignment of jobs to processors under heavy traffic

Abstract: The paper is concerned with the optimal control of the assignment of jobs from several arriving random streams to one of a bank of processors. Owing to the difficulty of the general problem, a heav...

Local Spherical Contact Distribution Function And Local Mean Densities For Inhomogeneous Random Sets

Abstract: In this paper we provide definitions for the local mean volume and mean surface densities of an inhomogeneous random closed set A theorem which relates the local spherical contact distribution function with the local surface and volume density is proven. Sufficient conditions on the regularity of the random set involved to satisfy the assumptions of the theorem are provided, based on Coarea Formula. These conditions are satisfied by a wide class of inhomogeneous random sets, relevant for applications, like some kinds of Boolean Models, for which explicit expressions for the local volume and surface densities are also provided

The Interarrival Hyperexponential Queues:Hk/M/c/N With Balking And Reneging

Abstract: This paper treats the analytical solution of the truncated interarrival hyperexponential queue. Hk/M/c/N with balking and reneging for general values of k,c and N. The discipline considered here is FIFO. Some previously published results are shown to be special cases of the present results

Sharp Laplace Asymptotics for a Hyperbolic SPDE

Abstract: In this paper we prove a precise asymptotic estimate for Laplace type functionals of a certain class of hyperbolic SPDEs. We use a large deviation principle, the stochastic Taylor’s expansion introduced by Azencott, and some methods of representation and estimation for our stochastic partial differential equation.

Sur la récurrence positive du mouvement Brownien réflechi dans l'orthant positif de Rn

Abstract: This paper is concerned with positive recurrence of a semimartingale reflecting brownian motion (SRBM) associated with the data (θ, R, S, Δ). We study here the particular case where S is a positive orthant of R 3 , R is a 3 × 3 P-matrix, θ and A are respectively the drift and a covariance matrix of the brownian motion. We use a geometrical approach to characterize the drift θ ∈ R 3 such that the SRBM is positive recurrent. In the two dimensional case, we recover the necessary and sufficient conditions for the positive recurrence. Some explicit necessary conditions are derived for the higher-dimensional case. Dans cet article nous etudions la recurrence positive d'un SRBM (Semi martingale Reflecting Brownian Motion) associe aux donnees (θ, R, S, Δ). Avec S est l'orthant positif de R 3 , R la matrice de regulation, θ et Δ sont respectivement le drift et la matrice de covariance du mouvement brownien. Nous utilisons une approche geometrique pour caracteriser les drifts' θ ∈ R 3 pour lesquels ce processus est positivement recurrent.

Regularity properties of transition probabilities in infinite dimensions

Abstract: In a Hilbert space H we consider a process X solution of a semilinear stochastic differential equation, driven by a Wiener process. We prove that, under appropriate conditions, the transition probabilities of X are absolutely continuous with respect to a properly chosen gaussian measure μ in H, and the corresponding densities belong to some Wiener-Sobolev spaces over (H,μ). In the linear caseX is a nonsymmetric Ornstein-Uhlenbeck process, with possibly degenerate diffusion coefficient. The general case is treated by the Girsanov. Theorem and the Malliavin calculus. Examples and applications to stochastic partial differential equations are given

Integration stochastique multivoque et inclusions differentielles stochastiques

Abstract: We will present several results concerning multivalued stochastic integration and existence of solutions for some stochastic differential inclusion under certain Lipschitz type conditions

On the -distance between semimartingales reflecting in different domains>

Abstract: Let D, D′ be either convex domains in or domains satisfying the conditions (A) and (B) considered by Lions and Sznitman and by Saisho. For given semimartingales Y, Y′ the -distance between solutions of the Skorokhod problem associated with Y, D and Y′, D′ is estimated. Applications to stability problems for stochastic equations with reflecting boundary conditions and to numerical approximation of solutions by the projection scheme and by the discrete penalization scheme are discussed in detail.

Equivalence Of Banach Space-Valued Ornstein-Uhlenbeck Processes

Abstract: Let μ A be the law of the Ornstein-Uhlenbeck process that solves the equation almost surely, where A is a generator of a C 0-semigroup on a Banach space B and W( t )t≥0, is a cylindrical Wiener process on a Hilbert subspace H of B. For two ‘drift’ operators A 0 and A 1 we study sufficient and necessary conditions for the measures and to be equivalent

Stratonovich Anticipative Stochastic Differential Equations In The Plane

Abstract: We apply the techniques of the quasi-sure analysis to prove that the stochastic differential equation in the plane with smooth and nondegenerate initial condition can be solved

On consumption/investment problems with long-term time-average utilities

Abstract: We study consumption/investment problems with long-term time-average utilities. The associated Hamilton-Jacobi-Bellman equation can be solved under some regularity conditions of utility rate function, and the optimal portfolio and consumption-rates are exhibited in explicit forms. An application to the optimization problem with finite horizon is also given

Predicting integrals of diffusion processes with unknown diffusion parameters

Abstract: Consider predicting the integral of a diffusion process Z in a bounded interval A given a set of observations , where is a dense triangular array of points (the step of discretization tends to zero as n increases) in the bounded interval. We predict using the conditional expectation of the integral of the diffusion process, the optimal predictor in terms of minimizing the mean squared error, given the observed values of the process. We present in this paper an easily computed approximation to the optimal predictor and an approximation to the standard error in the prediction, assuming the diffusion parameters are unknown. The approximations obtained in this paper are asymptotically optimal and they are just simple functions of the observed diffusion values

Double points of the Brownian sheet in Rd and the geometry.of the parameter space

Abstract: For the Brownian sheet W with values in R dthe set of double points , i.e., points for which Ws = Wu is investigated, in terms of the corresponding self intersection local time. Its existence is seen to depend sensitively upon the geometric constellation of the compact sets A B : . We suppose that A and B intersect at exactly one point p, and can be separated locally by an axial parallel line. We further assume that their boundaries in a vicinity of p are given by power type functions. We compute the critical dimension below which self intersection local time exists and describe it in terms of the powers of the boundary curves of A and B.

Almost sure optimality and optimality in probability for stochastic linear-quadratic regulator with partial information

Abstract: Partially observed linear-quadratic regulator is considered over an infinite time horizon. A limiting per unit time inequality is proved for the random difference between the cost corresponding to the feedback control based on Kalman filter estimates and the cost corresponding to an alternative control. Under suitable assumptions of admissibility for a control, it is shown that the feedback control mentioned above is asymptotically optimal almost surely and in probability

One-dimensional scale invariant diffusions

Abstract: Examples of one-dimensional scale invariant diffusions include Brownian motion and Bessel processes. After multiplication by – 1 if necessary, such a diffusion has state space or . Barring singular cases, the diffusion in is shown to be a positive multiple of a Bessel process, up to the first hit of . In it is composed of two pieces “appropriately joined”: the piece in is a positive multiple of a Bessel process and the piece in is a negative multiple of a Bessel process. The key feature here is that the multipliers need not coincide in absolute value, and the parameters of the Bessel processes associated with each piece must have common value in (0,2). In the diffusion is either a positive multiple of a Bessel process absorbed at or such without absorption. We also describe the possibilities in singular cases.