Showing papers in "Stochastics and Stochastics Reports in 1999"
TL;DR: This work analyses the convergence of stochastic algorithms with Markovian noise when the ergodicity of the Markov chain governing the noise rapidly decreases as the control parameter tends to infinity and provides sufficient condition which ensure convergence.
Abstract: We analyse the convergence of stochastic algorithms with Markovian noise when the ergodicity of the Markov chain governing the noise rapidly decreases as the control parameter tends to infinity. In such a case, there may be a positive probability of divergence of the algorithm in the classic Robbins-Monro form. We provide sufficient condition which ensure convergence. Moreover, we analyse the asymptotic behaviour of these algorithms and state a diffusion approximation theorem
165 citations
TL;DR: In this paper, the authors show that their earlier results in [9] can be extended to Hilbert spaces and give examples of backward stochastic partial differential equations which can be solved w...
Abstract: The aim of this paper is to show that our earlier results in [9] can be extended to Hilbert spaces. We then give examples of backward stochastic partial differential equations which can be solved w...
60 citations
TL;DR: In this article, the authors consider the singular stochastic control problem of a linear, time-homogeneous and regular diffusion process and show that the smooth fit principle can be interpreted as an ordinary first order necessary condition for optimality.
Abstract: We consider the singular stochastic control problem of a linear, time-homogeneous and regular diffusion process. By relying on a combination of stochastic calculus the classical theory of diffusions, and ordinary nonlinear programming techniques, we find the optimal policies and their value functions in the three most common cases appearing in the applications of singular controls. Especially we demonstrate that the smooth fit principle can be interpreted as an ordinary first order necessary condition for optimality
59 citations
TL;DR: In this paper, a reformulation of the classical Wiener-Hopf factorization for random walks is given; this is applied to the study of the asymptotic behaviour of the ladder variables, the distribution of the maximum and the renewal mass function in the bivariate renewal process of ladder times and heights.
Abstract: A reformulation of the classical Wiener-Hopf factorization for random walks is given; this is applied to the study of the asymptotic behaviour of the ladder variables, the distribution of the maximum and the renewal mass function in the bivariate renewal process of ladder times and heights
53 citations
TL;DR: In this article, both Ito's stochastic differential equations, as well as equations driven by semimartingales, with non degenerate diffusion coefficient, are considered.
Abstract: Both Ito's stochastic differential equations, as well as equations driven by semimartingales, with non degenerate diffusion coefficient, are considered. Multidimensional pathwise uniqueness and non...
42 citations
TL;DR: In this paper, a second-order stochastic parabolic equation with zero Dirichlet boundary conditions is considered in a sufficiently smooth bounded domain, and existence, uniqueness, and regularity of the solution are established without assuming any compatibility relations.
Abstract: A second-order stochastic parabolic equation with zero Dirichlet boundary conditions is considered in a sufficiently smooth bounded domain. Existence, uniqueness, and regularity of the solution are established without assuming any compatibility relations. To control the solution near the boundary of the region, special Sobolev-type spaces with weights are introduced. To illustrate the results, two examples are considered: general linear equation with finite-dimensional noise and equation on a line segment, driven by space-time white noise.
40 citations
TL;DR: In this paper, the hitting probability of the d-dimensional fractional Brownian motion with index was studied and sufficient conditions for a compact set to be a polar set for X(t) were proved.
Abstract: Let be the d-dimensional fractional Brownian motion with index . The upper and lower bounds on the hitting probabilities of X(t) are obtained. Sufficient conditions for a compact set to be a polar set for X(t) are proved. It is also proved that if N≤αd, then for any compact set and if N<αd, then for any compact set where B(R d ) denotes the Borel σ-algebra in R d , and where dim and Dim are HausdorfT dimension and packing dimension respectively
32 citations
TL;DR: In this article, the Dirichlet form associated with the Brownian motion on simple nested fractals with an integral Lipschitz space was identified, which generalises such an identification on the Sierpinnski gasket.
Abstract: In this paper we identify the domain of the Dirichlet form associated with the Brownian motion on simple nested fractals with an integral Lipschitz space. This result generalises such an identification on the Sierpinnski gasket, carried out by Jonsson in [9]
31 citations
Journal Article•
28 citations
TL;DR: In this paper, the authors considered a backward stochastic differential equation, where W is a d-dimensional Brownian motion and the symmetric local time of Fat the level a, v is a signed measure on is a -measurable random variable in and is an adapted map from to.
Abstract: In this paper we deal with the following backward stochastic differential equation: where W is a d-dimensional Brownian motion is the symmetric local time of Fat the level a, v is a signed measure on is a -measurable random variable in and is an adapted map from to . If h is continuous with linear growth, we show the existence of a solution (Y,Z) for this backward equation. Some applications of this result, in connection with partial differential equations, and with linear quadratic stochastic control problem, are also given
28 citations
TL;DR: In this article, the existence of solutions to the Bellman equation corresponding to risk sensitive control of partially observed discrete time Markov processes is shown; this in turn leads to optimal strategies, and the method used in the paper is based on discounted risk sensitive approximation.
Abstract: In this paper existence of solutions to the Bellman equation corresponding to risk sensitive control of partially observed discrete time Markov processes is shown; this in turn leads to the existence of optimal strategies. The method used in the paper is based on discounted risk sensitive approximation
TL;DR: In this paper, the authors considered 1-dimensional diffusions with jumps and showed that the transition semigroup corresponding to the diffusion is strong Feller and irreducible under some smoothness and growth conditions.
Abstract: We consider 1 -dimensional diffusions with jumps. It is shown that the transition semigroup corresponding to the diffusion is strong Feller and irreducible under some smoothness and growth conditio...
TL;DR: In this paper, it was shown that the random attractor associated to the stochastic Navier-Stokes equation has a finite Hausdorff dimension, and that the asymptotic behavior of the random dynamical system is governed by a finite number of degrees of freedom.
Abstract: The recent theory of random attractors is becoming very useful for the study of the asymptotic behaviour of dissipative random dynamical systems. A random attractor is a random invariant compact set which attracts every trajectory as time becomes infinite. Moreover, it can be proved that, in some cases, the random attractor has finite HausdorfT dimension. It seems that the asymptotic behaviour of the random dynamical system is governed by a finite number of degrees of freedom. In this work we prove a result in this direction: we generalize a result from the (deterministic) theory of dissipative dynamical systems known in the literature as determining modes. The result is applied to Navier – Stokes equation and a problem of reaction-diffusion type, both with additive white noise. We finally prove that the random attractor associated to the stochastic Navier – Stokes equation has finite Hausdorff dimension
TL;DR: In this paper, the authors prove existence and uniqueness of the invariant measure and convergence to equilibrium for a class of stochastic reaction-diffusion systems defined in bounded domains of Rd with the reaction term having polynomial growth.
Abstract: We prove existence and uniqueness of the invariant measure and convergence to equilibrium for a class of stochastic reaction-diffusion systems defined in bounded domains of Rd with the reaction term having polynomial growth
TL;DR: In this article, existence and uniqueness theorems for a class of stochastic partial differential equations with nonlinearities of polynomial growth in the case of one space dimension were proved.
Abstract: We prove existence and uniqueness theorems for a class of stochastic partial differential equations with nonlinearities of polynomial growth in the case of one space-dimension
TL;DR: In this article, a heavy traffic analysis of the optimal control of a classical manufacturing and inventory process, called Assemble-to-Order, is presented, where demand consists of one or more final products, each requiring either one or several of each of various part types.
Abstract: We do a heavy traffic analysis of the optimal control of a classical manufacturing and inventory process, called Assemble-to-Order. Demand consists of one or more final products, each requiring either one or several of each of various part types. The intervals between demands are random and might occur either singly or in batches. The part types are produced by dedicated processors, with random production times. Unneeded parts of each type are stored in a finite buffer. When a demand for a final product arrives, if all needed parts are available, the product is assembled and delivered. Otherwise, the demand can either be lost or backlogged. Control is exercised by idling the processors or possibly via external supply sources. Thorough analyses of a great variety of problem formulations of the optimal control problem are given. The general limit problem is what is called a singular control problem, or a combination of the singular and impulsive control problems. New issues in heavy traffic analysis arise. ...
TL;DR: In this paper, the authors extend the results of [9] on the stochastic disturbance attenuation problem to a wider class of systems and derive an H ∞-type theory for such systems.
Abstract: In this paper we extend the results of [9] on the stochastic disturbance attenuation problem to a wider class of stochastic systems. We consider time-invariant stochastic linear plants which are controlled by dynamic output feedback and subjected to both deterministic and stochastic parameter perturbations. The aim is to develop an H ∞-type theory for such systems. The Ito equations considered in [9] contained two (not necessarily independent) scalar Wiener processes, one for the state dependent noise term and one for the input dependent noise term. In this paper the two scalar Wiener processes are replaced by (not necessarily independent) vector valued Wiener processes. For this wider class of systems a bounded real lemma is derived which provides the basis for an LMI approach towards the stochastic disturbance attenuation problem. Necessary and sufficient conditions are derived for the existence of a stabilizing controller reducing the norm of the closed loop perturbation operator to a level below a giv...
TL;DR: In this article, the authors consider the almost sure asymptotic stability of the trajectory of a stochastic functional differential system with infinite memory in which the noise term does not depend on the history of the history.
Abstract: We consider the almost sure asymptotic stability of the trajectory of a stochastic functional differential system with infinite memory in which the noise term does not depend on the history of the ...
TL;DR: In this paper, a necessary and sufficient condition for the mean square stabilization of time-varying linear systems of ordinary differential equations by zero mean real noise is obtained, where the noise sources are generated either by Ornstein-Uhlenbek process or telegraphic process.
Abstract: A necessary and sufficient condition for the mean square stabilization of time-varying linear systems of ordinary differential equations by zero mean real noise is obtained. The noise sources are generated either by Ornstein-Uhlenbek process or telegraphic process
TL;DR: In this paper, the authors prove the existence of a wide collection of finite sets of functionals that completely determine the long-term behavior of solutions to 2D Navier-Stokes equations with random initial data and excited by an additive white noise.
Abstract: We prove the existence of a wide collection of finite sets of functionals that completely determine the long-term behaviour of solutions to 2D Navier-Stokes equations with random initial data and excited by an additive white noise. This collection contains finite sets of determining modes, nodes and local volume averages. We also show that determining functionals can be defined on only one of the components of the velocity vector. To characterize sets of determining functionals we invoke the concept of completeness defect. Our method is general and can be applied to other dissipative evolutionary infinite-dimensional equations driven by additive stochastic perturbations
TL;DR: In this article, the authors established conditions under which the maxingale problem defined in part I of the work has a unique solution and combined the uniqueness results and limit theorems of part I to obtain results on the large deviation principle for processes in a Skorohod space.
Abstract: We establish conditions under which the maxingale problem defined in part I of the work has a unique solution. We next combine the uniqueness results and limit theorems of part I to obtain results on the large deviation principle for processes in a Skorohod space
TL;DR: In this paper, the vanishing discount limit of the value function for ergodic control of partially observed diffusions is shown to exist as the limit for the discounted version of the discounted problem, and sufficient conditions for optimality are derived in the framework of the "martingale formulation".
Abstract: The value function for ergodic control of a class of partially observed diffusions is shown to exist as the 'vanishing discount' limit of the value function for the discounted problem. A verification theorem giving sufficient conditions for optimality is derived in the framework of the 'martingale formulation'.
TL;DR: In this paper, the authors extend the Freidlin-Wentzell large deviations principle in Besov-Orlicz space for diffusion processes and prove a Strassen's law of the iterated logarithm.
Abstract: Dans cet article nous demontrons le principe de grandes deviations de Freidlin et Wentzeii pour les diffusions suivant la topologie de l'espace de Besov-Orlicz. Nous etendons ainsi le resultat de Ben Arous et Ledoux [3] concernant le cas holderien et celui de Roynetie [16] tiaitant le cas du mouvement brownien en norrne de Besov Comme application on etablit la loi fonctionnelle du Logarithme itere de Strassen pour l'aire de Levy en norme de Besov-Orlicz. Cette loi est plus fine que celle obtenue par N'zi et Eddahbi [14] avec la norme Holderienne In this paper, we prove a Freidlin-Wentzell large deviations principle in Besov-Orlicz space for diffusion processes. Our result is an extension of that of Ben-Arous and Ledoux [3] who have treated the Holder norm case and that of Roynette [16] who have considered Brownian motion in Besov space. As an application, we establish a Strassen's law of the iterated logarithm for Levy's area process. This law is more sharper than the one in Holder norm obtained by N'zi a...
TL;DR: In this article, the authors consider a one-dimensional stochastic differential equation driven by a compensated Poisson measure and prove that under a strong non-degeneracy condition, for each t ≥ 0, the law of is bounded below by a measure that admits a strictly positive continuous density with respect to the Lebesgue measure.
Abstract: We consider a one-dimensional stochastic differential equation driven by a compensated Poisson measure. We assume that this equation admits a unique solution We prove that under a strong non-degeneracy condition, for each t ≥ 0, the law of is bounded below by a measure that admits a strictly positive continuous density with respect to the Lebesgue measure on . To this aim, we develop Bismut's approach of the Malliavin calculus for Poisson functionals
TL;DR: In this article, the stability of the solution of a stochastic partial functional -differential equation that describes the evolution of a population is shown, and a comparison between solutions of time-delayed problems and solutions of problems without time delay is given.
Abstract: Stability of the solution of a stochastic partial functional - differential equation that describes the evolution of a population is shown. A comparison between solutions of time-delayed problems and solutions of problems without time delay is given
TL;DR: In this article, the authors studied the asymptotic behavior of oscillatory integrals on the Wiener space whose phase function is a non-degenerate double stochastic integral.
Abstract: This paper is devoted to study the asymptotic behavior of oscillatory integrals on the Wiener space whose phase function is a non-degenerate double stochastic integral. An expansion for these integrals is obtained in terms of the derivative of F at the origin using its Wiener chas expansion
TL;DR: In this paper, the Holder condition for the local time of a real-valued symmetric stable process of index β, 1 0 and p < ∞ has been studied and the random function satisfies a Holder condition in the Lp norm, with the modulus of smoothness.
Abstract: Let be a real-valued symmetric stable process of index β,1 0 and p < ∞, the random function satisfies a.s a Holder condition in the Lp norm, with the modulus of smoothness In this paper the Holder condition for the local time of the symmetric stable process of index 1 < β ≤ 2 in th...
TL;DR: In this paper, the authors present sufficient conditions for sample path large deviation principles to be extended to finer topologies, such as Lipschitz spaces and Orlicz functional topologies.
Abstract: In this paper we present sufficient conditions for sample path large deviation principles to be extended to finer topologies. We consider extensions of the uniform topology by Orlicz functional and we consider Lipschitz spaces: the former are concerned with cumulative path behavior while the latter are more sensitive to extremes in local variation. We also consider sample paths indexed by the half line, where the usual projective limit topologies are not strong enough for many applications. We introduce and apply a new technique extending large deviation principles to finer topologies. We show how to apply the results to obtain large deviations for weighted statistics, to improve Schilder's theorem as well as to obtain large deviations in queueing theory
TL;DR: In this paper, a one-dimensional stochastic differential equation (S) without drift but with time-dependent diffusion coefficient is considered, and the time change method is applied to obtain weak solutions.
Abstract: Consider a one-dimensional stochastic differential equation (S) without drift but with time-dependent diffusion coefficient. To obtain weak solutions, the time change method is applied: An increasing process is looked for such that a given Brownian motion, distorted by this process in its time argument, turns out to be a solution of (S). For this purpose, a time change equation(TC) has to be solved. We present one-to-one relations between (S) and (TC) concerning existence and uniqueness. Contrary to SDEs, pathwise uniqueness for (TC) coincides with that in law. By solving (TC), solutions of (S) are constructed under weak conditions admitting degenerate diffusion. The results improve those of Senf [18] and Rozkosz and Slomin nski [17]. Apart from degenerate diffusion, the main difference is that monotone approximation for the solutions of (TC), in contrast to weak convergence, is systematically exploited. As a consequence, the constructed solutions can be identified as pure so that they have the representa...
TL;DR: In this paper, the problem of mean-variance hedging by continuous trading of futures contracts is discussed under stochastic interest-rate setting, and the L 2-optimal strategies are obtained.
Abstract: The problem of mean-variance hedging by continuous trading of futures contracts is discussed under stochastic interest-rate setting. In this situation, a hedger owes not only the risk caused by the randomness of a payoff but also the risk caused by random interest-rate, and tries to control both of them by futures trading. Therefore, the so-called “projection method” cannot be directly applied. By using the measure-change-technique via Feynman-Kac's formula, which Davis has suggested (1998), we shall observe that the problem can be modified to apply “the projection method” and the L 2-optimal strategies shall be obtained. Further, we will compute mean-variance-efficient strategies and frontiers; simple expressions are obtained even in our stochastic-interest-rate setting, and the impact of “random interest-rate risk”is clarified