Showing papers in "Stochastic Analysis and Applications in 2003"
TL;DR: In this article, the local and global existence of solutions in the energy space H 1(R n ) for stochastic nonlinear Schrodinger equations with either additive or multiplicative noise was investigated.
Abstract: We investigate the local and global existence of solutions in the energy space H 1(R n ) for stochastic nonlinear Schrodinger equations with either additive or multiplicative noise. The noise is assumed to be white in time and correlated in the space variables.
186 citations
TL;DR: In this paper, a fundamental theory for NSDDEs with Markovian switching is established and some important properties of the solutions are discussed, e.g. boundedness and sta...
Abstract: Neutral stochastic differential delay equations (NSDDEs) have recently been studied intensively (see Kolmanovskii, V.B. and Nosov, V.R., Stability and Periodic Modes of Control Systems with Aftereffect; Nauka: Moscow, 1981 and Mao X., Stochastic Differential Equations and Their Applications; Horwood Pub.: Chichester, 1997). Given that many systems are often subject to component failures or repairs, changing subsystem interconnections and abrupt environmental disturbances etc., the structure and parameters of underlying NSDDEs may change abruptly. One way to model such abrupt changes is to use the continuous‐time Markov chains. As a result, the underlying NSDDEs become NSDDEs with Markovian switching which are hybrid systems. So far little is known about the NSDDEs with Markovian switching and the aim of this paper is to close this gap. In this paper we will not only establish a fundamental theory for such systems but also discuss some important properties of the solutions e.g. boundedness and sta...
138 citations
TL;DR: In this article, a set-valued stochastic integral with respect to a 1-dimensional Brownian motion is defined, and multivalued analogs to the theory of single-valued integrals are developed.
Abstract: We define a set-valued stochastic integral with respect to a 1-dimensional Brownian motion. The paper develops multivalued analogs to the theory of singlevalued stochastic integrals. It is expected that these results will be useful to study set-valued and fuzzy stochastic analysis.
75 citations
TL;DR: In this article, the authors consider optimal control problems where the state X(t) at time t of the system is given by a stochastic differential delay equation, and derive an associated (finite dimensional) Hamilton-Jacobi-Bellman equation for the value function of such problems.
Abstract: We consider optimal control problems where the state X(t) at time t of the system is given by a stochastic differential delay equation. The growth at time t not only depends on the present value X(t), but also on X(t-δ) and some sliding average of previous values. Moreover, this dependence may be nonlinear. Using the dynamic programming principle we derive an associated (finite dimensional) Hamilton-Jacobi-Bellman equation for the value function of such problems. This (finite dimensional) HJB equation has solutions if and only if the coefficients satisfy a particular system of first order PDEs. We introduce viscosity solutions for the type of HJB-equations that we consider, and prove that under certain conditions, the value function is the unique viscosity solution to the HJB-equation. We also give numerical examples for two cases where the HJB-equation reduces to a finite dimensional one.
55 citations
TL;DR: In this paper, the existence and stability problems associated with semilinear stochastic evolution equations with variable delay in infinite dimensions were considered and compared under a comparison principle under less restrictive hypothesis than the Lipschitz condition on the nonlinear terms.
Abstract: In this paper, we consider the existence and stability problems associated with semilinear stochastic evolution equations with variable delay in infinite dimensions. To be precise, we first study an existence result and then the exponential stability of a mild solution as well as asymptotic stability in probability of its sample paths. Such results are established employing a comparison principle under less restrictive hypothesis than the Lipschitz condition on the nonlinear terms. An application is included to illustrate the theory.
51 citations
TL;DR: In this article, the covariance factorization of an α-generalized random field having a dual is established, leading to a white-noise linear-filter representation, which reduces to the usual Markov representation in the ordinary case when α∈N and the variance operator of the dual random field is local.
Abstract: Using the theory of generalized random fields on fractional Sobolev spaces on bounded domains, and the concept of dual generalized random field, this paper introduces a class of random fields with fractional-order pure point spectra. The covariance factorization of an α-generalized random field having a dual is established, leading to a white-noise linear-filter representation, which reduces to the usual Markov representation in the ordinary case when α∈N and the covariance operator of the dual random field is local. Fractional-order differential models commonly arising from anomalous diffusion in disordered media can be studied within this framework.
48 citations
TL;DR: In this paper, the concept of contingent curvature of a set and contingent epi-Hessian of a functio is defined and it is shown that these first-order normal conditions are equivalent to second-order tangential conditions that we expect for invariance under stochastic differential equations for smooth subsets.
Abstract: Thanks to the Stroock and Varadhan “Support Theorem” and under convenient regularity assumptions, stochastic viability problems are equivalent to invariance problems for control systems (also called tychastic viability), as it has been singled out by Doss in 1977 for instance By the way, it is in this framework of invariance under control systems that problems of stochastic viability in mathematical finance are studied The Invariance Theorem for control systems characterizes invariance through first‐order tangential and/or normal conditions whereas the stochastic invariance theorem characterizes invariance under second‐order tangential conditions Doss's Theorem states that these first‐order normal conditions are equivalent to second‐order normal conditions that we expect for invariance under stochastic differential equations for smooth subsets We extend this result to any subset by defining in an adequate way the concept of contingent curvature of a set and contingent epi‐Hessian of a functio
48 citations
TL;DR: In this article, the authors studied a class of forward-backward stochastic differential equations with functional-type terminal conditions and proved the existence and uniqueness of the strong adapted solution in the usual sense.
Abstract: In this note we study a class of forward–backward stochastic differential equations (FBSDE for short) with functional-type terminal conditions. In the case when the time duration and the coefficients are “compatible” (e.g., the time duration is small), we prove the existence and uniqueness of the strong adapted solution in the usual sense. In the general case we introduce a notion of weak solution for such FBSDEs, as well as two notions of uniqueness. We prove the existence of the weak solution under mild conditions, and we prove that the Yamada–Watanabe Theorem, that is, pathwise uniqueness implies uniqueness in law, as well as the Principle of Causality also hold in this context.
43 citations
TL;DR: In this paper, an algorithm that provides a necessary and sufficient condition for exponential stabilization in the mean square is derived, based on the solutions of matrix Riccati-like algebraic equations.
Abstract: In active Fault Tolerant Control Systems (FTCSs), a Fault Detection and Diagnosis (FDD) algorithm is used to monitor system performance, detect the occurrence of faults and estimate post-fault system parameters. Usually, estimation techniques employed by the FDD algorithm will provide system parameters with some inaccuracies. As a result, the system to be controlled will include parameter uncertainties. In this paper, exponential stabilization of FTCSs for systems with time-varying unknown-but-bounded parameter uncertainties is considered. In particular, an algorithm that provides a necessary and sufficient condition for exponential stabilization in the mean square is derived. The stabilizing controller is calculated, based on the solutions of matrix Riccati-like algebraic equations. Several interesting special cases are derived from the general result. A numerical example is presented to demonstrate the theoretical developments.
41 citations
TL;DR: In this paper, the stochastic stabilization problem for a class of linear discrete time-delay systems with Markovian jump parameters is investigated, and sufficient conditions are proposed to solve the above problems, which are in terms of a set of solutions of coupled matrix inequalities.
Abstract: In this paper, we investigate the stochastic stabilization problem for a class of linear discrete time-delay systems with Markovian jump parameters. The jump parameters considered here is modeled by a discrete-time Markov chain. Our attention is focused on the design of linear state feedback memoryless controller such that stochastic stability of the resulting closed-loop system is guaranteed when the system under consideration is either with or without parameter uncertainties. Sufficient conditions are proposed to solve the above problems, which are in terms of a set of solutions of coupled matrix inequalities.
32 citations
TL;DR: In this article, a purely probabilistic approach was used to study forward-backward differential equations with Poisson jumps with stopping time as termination, and the existence and uniqueness results of solutions were obtained under weak monotonicity conditions and Lipschitz conditions.
Abstract: In this paper, we use a purely probabilistic approach to study forward‐backward differential equations with Poisson jumps with stopping time as termination. Under some weak monotonicity conditions and Lipschitz conditions, the existence and uniqueness results of solutions are obtained, it may be served as the generalized results contrast to FBDE with Brownian motion. We also derive the convergence theorem of the solutions.
TL;DR: In this article, the pathwise asymptotic stability of solutions to stochastic partial differential equations is studied and sufficient conditions for almost sure stability with a non-exponential decay rate are given.
Abstract: Some results on the pathwise asymptotic stability of solutions to stochastic partial differential equations are proved. Special attention is paid in proving sufficient conditions ensuring almost sure asymptotic stability with a non-exponential decay rate. The situation containing some hereditary characteristics is also treated. The results are illustrated with several examples.
TL;DR: In this article, the authors considered a two-phase queueing system with server vacations and Bernoulli feedback and derived a relationship between the generating functions for the system size at various embedded epochs.
Abstract: We consider a two‐phase queueing system with server vacations and Bernoulli feedback. Customers arrive at the system according to a Poisson process and receive batch service in the first phase followed by individual services in the second phase. Each customer who completes the individual service returns to the tail of the second phase service queue with probability 1 − σ. If the system becomes empty at the moment of the completion of the second phase services, the server takes vacations until he finds customers. This type of queueing problem can be easily found in computer and telecommunication systems. By deriving a relationship between the generating functions for the system size at various embedded epochs, we obtain the system size distribution at an arbitrary time. The exhaustive and gated cases for the batch service are considered.
TL;DR: In this paper, constrained continuous-time Markov control processes with a denumerable state space and unbounded reward/cost and transition rates are studied, and conditions that ensure the existence of constrained-optimal policies are given.
Abstract: In this paper we study constrained continuous-time Markov control processes with a denumerable state space and unbounded reward/cost and transition rates. The criterion to be maximized is an expected discounted reward, and the constraint is imposed on an expected discounted cost. We give conditions that ensure the existence of constrained-optimal policies. We also show that a constrained-optimal policy may be a stationary policy or a randomized stationary policy that randomizes between two stationary policies which differ in at most one state. Our results are illustrated with a controlled queueing system.
TL;DR: In this article, the authors show that slow variation at the origin of (an absolutely continuous) distribution function is incompatible with self-decomposability and this is shown in three examples.
Abstract: Stationary (limiting) distributions of shot noise processes, with exponential response functions, form a large subclass of positive selfdecomposable distributions that we illustrate by many examples. These shot noise distributions are described among selfdecomposable ones via the regular variation at zero of their distribution functions. However, slow variation at the origin of (an absolutely continuous) distribution function is incompatible with selfdecomposability and this is shown in three examples.
TL;DR: In this article, the authors consider the stability of mild solutions of stochastic delay evolution equations of monotone type and propose an Ito-type inequality to study the stability in the p-th moment.
Abstract: In this paper, we consider the exponentially asymptotic stability of the mild solutions of semilinear stochastic delay evolution equations of monotone type. An Ito-type inequality is our main tool to study the stability in the p-th moment and almost sure sample-path stability of the mild solutions. At last, we give some examples to illustrate the applications of the theorems.
TL;DR: In this article, a class of bilinear time series models with time varying coefficients is considered and a necessary and sufficient condition for the existence and uniqueness of a solution with bounded first and second order moments (BFSM) is provided.
Abstract: In this paper, a class of bilinear time series models with time varying coefficients is considered. In this nonstationary and nonlinear framework, our aim is to study the structure of usual time series analysis tools, in particular the sample autocovariance function which has been developed for analyzing stationary linear time series. We use appropriately defined Markovian representations to derive a necessary and sufficient condition for the existence and uniqueness of a solution with bounded first and second order moments (BFSM). A more explicit sufficient condition for the existence of a BFSM solution is provided. An explicit expression of the autocovariance function is obtained. The existence of a weak time-varying ARMA representation of the bilinear model with time varying coefficients is shown. We also discuss the existence of higher order moments. Several subclasses of the model are shown to be quasi-stationary. Under this assumption of quasi-stationarity, the asymptotic distributions of the sample...
TL;DR: In this article, an exponential formula for the reachable sets of quantum stochastic differential inclusions (QSDI) with convex values was established, which partially relies on an auxilliary result concerning the density, in the topology of the locally convex space of solutions, of the set of trajectories whose matrix elements are continuously differentiable.
Abstract: We establish an exponential formula for the reachable sets of quantum stochastic differential inclusions (QSDI) which are locally Lipschitzian with convex values. Our main results partially rely on an auxilliary result concerning the density, in the topology of the locally convex space of solutions, of the set of trajectories whose matrix elements are continuously differentiable. By applying the exponential formula, we obtain results concerning convergence of the discrete approximations of the reachable set of the QSDI. This extends similar results of Wolenski[20] for classical differential inclusions to the present noncommutative quantum setting.
TL;DR: In this paper, the authors considered the class of stochastic linear Volterra equations of convolution type defined by fractional integration kernels g ρ (t)={ 1 Γ(ρ) }t ρ−1, ρ∈(0,2) using an explicit formula for the scalar resolvent function.
Abstract: We consider the class of stochastic linear Volterra equations of convolution type defined by fractional integration kernels g ρ (t)={ 1 Γ(ρ) }t ρ−1, ρ∈(0,2) Using an explicit formula for the scalar resolvent function, we establish the basic properties of the stochastic convolution process W S Our formulas are given in terms of the Mittag-Leffler's function
TL;DR: In this article, a bisexual Galton-Watson branching process (BGWP) was introduced, in which the offspring probability distribution is different in each generation, and sufficient and necessary conditions for its almost sure extinction were provided.
Abstract: In this paper we introduce a bisexual Galton‐Watson branching process (BGWP) in which the offspring probability distribution is different in each generation. We obtain some relations among the probability generating functions (pgf) involved in the model and, making use of mean growth rates and fractional linear functions (flf), we provide sufficient and necessary conditions for its almost sure extinction.
TL;DR: In this article, the expected density of real zeros on (−1/p, 1/p) has the limit p/(1−p 2 x 2), where p 2j =Var(Δ j ).
Abstract: Let be a random algebraic polynomial where the coefficients A 0,A 1,… form a sequence of centered Gaussian random variables. Moreover, assume that the increments Δ j =A j −A j−1,j=0,1,2,… are independent, A −1=0. The coefficients A 0, A 1, … A n can be considered as n consecutive observations of a Brownian motion. We provide an explicit formula for the expected density of the number of real zeros of Q n (x). We observe that the expected density of real zeros on (−1/p,1/p) has the limit p/(1−p 2 x 2), n→∞ where p 2j =Var(Δ j ), and we obtain the asymptotic behaviour of the expected number of real zeros for the case that p=1, which exhibits new features in the study of random algebraic polynomials.
TL;DR: In this paper, the existence and uniqueness of mild solutions for a class of first-order abstract stochastic Sobolev-type integro-differential equations in a real separable Hilbert space were established.
Abstract: We establish the global existence and uniqueness of mild solutions for a class of first‐order abstract stochastic Sobolev‐type integro‐differential equations in a real separable Hilbert space in which we allow the nonlinearities at a given time t to depend not only on the state of the solution at time, t, but also on the corresponding probability distribution at time t. Results concerning the continuous dependence of solutions on the initial data and almost sure exponential stability, as well as an extension of the existence result to the case in which the classical initial condition is replaced by a so‐called nonlocal initial condition, are also discussed. Finally, an example is provided to illustrate the applicability of the general theory.
TL;DR: In this paper, the error estimates involved in the solution of a discrete approximation of a quantum stochastic differential inclusion (QSDI) were investigated and the main results rely on certain properties of the averaged modulus of continuity for multivalued sesquilinear forms associated with QSDI.
Abstract: This paper is concerned with the error estimates involved in the solution of a discrete approximation of a quantum stochastic differential inclusion (QSDI). Our main results rely on certain properties of the averaged modulus of continuity for multivalued sesquilinear forms associated with QSDI. We obtained results concerning the estimates of the Hausdorff distance between the set of solutions of the QSDI and the set of solutions of its discrete approximation. This extends the results of Dontchev and Farkhi Dontchev, A.L.; Farkhi, E.M. (Error estimates for discre‐ tized differential inclusions. Computing 1989, 41, 349–358) concerning classical differential inclusions to the present noncommutative quantum setting involving inclusions in certain locally convex space.
TL;DR: It is proved that the stationary distribution of the GI X /M Y /1 queue is asymptotically geometric and the asymPTotic geometric parameter is the same as the geometric parameter of the upper bound.
Abstract: For the GI X /M/1 queue, it has been recently proved that there exist geometric distributions that are stochastic lower and upper bounds for the stationary distribution of the embedded Markov chain at arrival epochs. In this note we observe that this is also true for the GI X /M Y /1 queue. Moreover, we prove that the stationary distribution of its embedded Markov chain is asymptotically geometric. It is noteworthy that the asymptotic geometric parameter is the same as the geometric parameter of the upper bound. This fact justifies previous numerical findings about the quality of the bounds.
TL;DR: In this article, the optimal consumption and investment problem for a large investor, who possesses information about the terminal values of the components of the Brownian motion, possibly distorted by "noise", is examined.
Abstract: This paper examines the optimal consumption and investment problem for a ‘large investor’, who possesses information about the terminal values of the components of the Brownian motion, possibly distorted by ‘noise’. Existence of optimal policies is established using martingale and duality techniques under general assumptions on the securities’ price process and the investor's preferences. Explicit solutions are provided for specific cases involving an agent with logarithmic utilities.
TL;DR: For weakly convex and concave random maps with position dependent probabilities, this article showed sufficient conditions for the existence of absolutely continuous invariant measures for weak convex random maps.
Abstract: A random map is a discrete‐time dynamical system in which one of a number of transformations is randomly selected and applied in each iteration of the process. In this paper, we study random maps with position dependent probabilities on the interval. Sufficient conditions for the existence of absolutely continuous invariant measures for weakly convex and concave random maps with position dependent probabilities is the main result of this note.
TL;DR: In this article, the backward equations of Markov skeleton processes were applied to GI/G/N queueing systems and the transient distribution of the length of the GI/g/n queueing system was obtained.
Abstract: In this paper, we first present the backward equations of Markov skeleton processes, which are then applied to GI/G/N queueing systems.Transient distribution of the length of GI/G/N queueing system is obtained.
TL;DR: In this paper, the first passage time of a one-dimensional diffusion process through a stochastic, as well as deterministic one-sided boundary is studied, generalizing the analogous results found by Peskir and Shiryaev for Brownian motion.
Abstract: Some problems about the asymptotics of the first-passage time of a one-dimensional diffusion process through a stochastic, as well as deterministic one-sided boundary are studied, generalizing the analogous results found by Peskir and Shiryaev for Brownian motion.
TL;DR: In this article, the authors considered four models for T based on the class of extreme value distributions (Gumbel, Fre´chet, Weibull and Pareto) and provided methods for estimating t α.
Abstract: When a new software is produced it is usually tested for failure several times in succession (whenever a failure is detected the software is rectified and tested again for failure). Suppose X 1,X 2,…,X k denote the times between failures. For the customer the main characteristic of interest is T=max(X 1,X 2,…,X k ). In particular, one would be interested in t α for which Pr{T≤t α }=1−α for small α. In this paper we consider four models for T based on the class of extreme value distributions (Gumbel, Fre´chet, Weibull and Pareto) and provide methods for estimating t α . In addition to numerical estimation of t α , we perform sensitivity analysis of t α with respect to the four models considered.
TL;DR: In this article, the strong consistency and asymptotic normality of the maximum likelihood estimator of a parameter appearing linearly in the drift coefficient of a partially observed nonlinear stochastic differential system was studied.
Abstract: We study the strong consistency and asymptotic normality of the maximum likelihood estimator of a parameter appearing linearly in the drift coefficient of a partially observed nonlinear stochastic differential system when the observed process is driven by fractional Brownian motion and unobserved (signal process) is driven by ordinary Brownian motion. We illustrate the results for a linear fractional stochastic system.