Showing papers in "Stochastic Analysis and Applications in 2000"
TL;DR: In this paper, the relative compactness of the stochastic particle systems related to the coagulation fragmentation equation is established and weak accumulation points are characterized as solutions, which imply a new existence theorem.
Abstract: This paper studies stochastic particle systems related to the coagulation fragmentation equation. For a certain class of unbounded coagulation kernels and fragmentation rates, relative compactness of the stochastic systems is established and weak accumulation points are characterized as solutions. These results imply a new existence theorem. Finally a simulation algorithm based on the particle systems is proposed
58 citations
TL;DR: In this paper, the stability of a stochastic integro-differential equation which is regarded as stochastically perturbed system of a given non-linear stable system is investigated.
Abstract: In this paper we shall investigate the stability of a stochastic integro–differential equation which is regarded as a stochastically perturbed system of a given non-linear stable system . Several sufficient criteria are obtained under which the system can tolerate the stochastic perturbation without losing the property of stability
36 citations
TL;DR: In this paper, the authors presented a model of single-machine scheduling problem, where the machine is failure-prone and subject to random breakdowns, and the processing time is a deterministic sequence that is randomly compressible, which may be from the introduction of new technology or addition of new equipment.
Abstract: This work presents a model of single-machine scheduling problem. The machine is failure-prone and subject to random breakdowns. The processing time is a deterministic sequence that is randomly compressible, which may be from the introduction of new technology or addition of new equipment. Taking into account the cost for the random breakdowns and the random compressible processing time, our objective is to find the optimal scheduling policy to minimize an objective function. Under simple conditions, it is shown that the optimal sequence possesses a V-shape property.
31 citations
TL;DR: In this article, a term structure model with lognormal type volatility structure is proposed to price and hedge caps, swaptions and other interest rate and currency derivatives including the Eurodollar futures contract, which requires integrability of one over zero coupon bond.
Abstract: A term structure model with lognormal type volatility structure is proposed. The Heath, Jarrow and Morton (HJM) framework, coupled with the theory of stochastic evolution equations in infinite dimensions, is used to show that the resulting instantaneous rates are well defined (they do not explode) and remain positive, contrary to those derived in [2]. They are also bounded from below and above by lognormal processes. The model can be used to price and hedge caps, swaptions and other interest rate and currency derivatives including the Eurodollar futures contract, which requires integrability of one over zero coupon bond. This extends results obtained by Sandmann and Sondermann in [22] and [23] for Markovian lognormal short rates to (non-Markovian) lognormal forward rates. We show also existence of invariant measures for the proposed term structure dynamics
31 citations
TL;DR: Two n species stochastic population models with periodic coefficients are studied in this paper, and sufficient conditions for the existence of asymptotically stable periodic solution process are obtained respectively.
Abstract: Two n species stochastic population models with periodic coefficients are studied. Some sufficient conditions for the existence of asymptotically stable periodic solution process are obtained respectively
27 citations
TL;DR: In this paper, the authors prove new results concerning the long-time behavior of random fields that are solutions in some generalized sense to a class of semilinear parabolic equations subjected to a homogeneous and multiplicative white noise.
Abstract: In this article we prove new results concerning the long-time behaviour of random fields that are solutions in some generalized sense to a class of semilinear parabolic equations subjected to a homogeneous and multiplicative white noise. Our main results state that these random fields eventually converge with probability. one to a global attractor represented by a single random variable whose properties we investigate in detail. We analyze the partial differential equations of this article in light Ito's stochastic calculus and thereby obtain stabilization and stability results which are substantially different from our earlier results concerning their interpretation in the sense of Stratonovitch. In particular, the asymptotic properties of the random fields that we investigate here exhibit no recurrence and oscillatory properties
26 citations
TL;DR: In this paper, a theory of combined probabilistic deterministic discretization with unilateral constraints is proposed for random variational inequalities and simple random elliptic boundary value problems with unilateral conditions, where randomness enters in the coefficient of the elliptic operator and in the right hand side of the p.d.
Abstract: The paper deals with a class of random variational inequalities and simple random elliptic boundary value problems with unilateral conditions. Here randomness enters in the coefficient of the elliptic operator and in the right hand side of the p.d.e. In addition to existence and uniqueness results a theory of combined probabilistic deterministic discretization is developped that includes nonconforming approxima¬tion of unilateral constraints. Without any regularity assumptions on the solution, norm convergence of the full approximation process is established. The theory is applied to a Helmholtz like elliptic equation with Signorini boundary conditions as a simple model problem, where Galerkin discretization is realized by finite element approximation
23 citations
TL;DR: In this paper, the stochastic difference equations with Volterra type linear and nonlinear main term are considered and conditions on a.s. boundedness of the solutions,asymptotic stability,exponential stability,and stability with a speed which differs from the exponential, are obtained.
Abstract: The stochastic difference equations with Volterra type linear and nonlinear main term are considered in the paper.Conditions on a.s. boundedness of the solutions,asymptotic stability,exponential stability,and stability with a speed which differs from the exponential,are obtained.A numerous examples are presented.
21 citations
TL;DR: In this paper, the stability of the solution of backward stochastic differ-ential equations under small perturbations of the underlying filtration has been studied from the mathematical point of view, but also for its possible applications in finance theory.
Abstract: This paper studies the stability of the solution of backward stochastic differential equations under small perturbations of the underlying filtration. The problem is interesting from the mathematical point of view, but also for its possible applications in Finance Theory, as it might represent a model to describe approximate information, inside or delayed information for an economic agent
20 citations
TL;DR: In this paper, the authors present a method of proof for solving non-linear optimal stopping problems based on time-change, and illustrate its use by considering several examples dealing with Brownian motion.
Abstract: Some non-linear optimal stopping problems can be solved explicitly by using a common method which is based on time-change. We describe this method and illustrate its use by considering several examples dealing with Brownian motion. In each of these examples we derive explicit formulas for the value function and display the optimal stopping time. The main emphasis of the paper is on the method of proof and its unifying scope.
20 citations
TL;DR: In this paper, the Gal'chuk-Davis method is extended to establish the comparision results of solutions to a class of stochastic systems of differential equations with diffusion coefficients defferent.
Abstract: Pathwise comparison of solutions to a class of stochastic systems of differential equations is proved which extends the existing result of Geiβ and Manthey. When the diffusion coefficients are defferent, the Gal’chuk-Davis method is extended to establish the comparision results. We illustrate our results with several examples some of which arise in stochastic finance theory
TL;DR: In this paper, a single machine scheduling problem with a common due date was considered, and sufficient conditions guaranteeing an optimal SEPT sequence were derived under exponential and normal processing times, and further results were obtained under normal and exponential processing times.
Abstract: This work is concerned with scheduling problems for a single machine. Taking earliness and tardiness of completion time and due–date value into consideration, the objective function with a common due date is considered. The processing time of each job is random. Sufficient conditions guaranteeing an optimal SEPT sequence are derived. Under exponential and normal processing times, further results are obtained
TL;DR: In this article, the authors give a family of convergence rates reached by the kernel estimator and show that these rates are minimax, and apply these results for specific classes of processes including Gaussian ones.
Abstract: In continuous time, rates of convergence of density estimators fluctuate with the nature of observed sample paths. In this paper, we give a family of rates reached by the kernel estimator and we show that these rates are minimax. Finally, we study applications of these results for specific classes of processes including the Gaussian ones
TL;DR: In this paper, the authors obtain sufficient conditions for transience and recurrence of multidimensional jump-diffusion processes, which are driven by Brownian motion and Poisson random measure.
Abstract: The purpose of this work is to obtain sufficient conditions for transience and recurrence of multidimensional jump–diffusion processes, which are driven by Brownian motion and Poisson random measure. The approach adopted here is to construct appropriate Lyapounov functions
TL;DR: In this article, the moments of the first crossing times over each boundary are shown to be the solutions of certain partial differential equations with suitable boundary conditions, and the results are illustrated graphically.
Abstract: Some problems of first-crossing times over two time-dependent boundaries for one-dimensional diffusion processes are considered. The moments of the first-crossing times over each boundary are shown to be the solutions of certain partial differential equations with suitable boundary conditions. For some examples where the boundaries are constant, the results are illustrated graphically.
TL;DR: In this article, it was shown that for a given dispersion coefficient σ(.), there will be a class drift coefficients b(.) are provided, provided a differential equation related to the one defining ρ as well as a simple reaction-diffusion equation strongly.
Abstract: This note is concerned with the study of explicit solutions to stochastic differential equations. Previously, Doss and Sussman showed that the unique strong solution to the scalar Ito equation X can be represented as a function ρ of a Brownian motion and an auxiliary stochastic process Yt determined, for every path of by ordinary differential equation (ODE). ρ itself is determined by a second differential equation. Now, it will be shown that X can be solved explicitly as with f(.) being a continuous real valued function, provided solves a differential equation related to the one defining ρ as well as a simple reaction-diffusion equation strongly. In particular, for a given dispersion coefficient σ(.), there will be a class drift coefficients b(.) are provided. The corresponding explicit solution xt for any given dispersion σ is also supplied
TL;DR: In this article, the authors employ the theory of dererministic ordinary differential inequalities together with the concept of vector Lyapunov-like functional to develop basic comparison theorems for system of partial differential equations of parabolic type under Markovian structural perturbations.
Abstract: In this paper we employ the theory of dererministic ordinary differential inequalities together with the concept of vector Lyapunov–like functional to develop basic comparison theorems for system of partial differential equations of parabolic type under Markovian structural perturbations.These results will be utilized to give sufficient conditions for the convergence and stability of the solution process of the system.We also characterize the effects of the random structural perturbations on the qualitative properties of such system. Moreover,the Lyapunov–like functional approach provides a mechanism to characterize the diffusion effects on the qualitative properties of the system.
TL;DR: In this article, a coupling method was used to estimate the Wasserstein distance between a simple point process and a Poisson process in terms of their compensators, which can be applied to establish accurate upper bound for the second part.
Abstract: We use coupling method to estimate a Wasserstein distance between a simple point process and a Poisson process in terms of their compensators. The Wasserstein distance of two probability measures Q1 and Q2 on configuration space of point processes on [0T] is defined as the sum of two parts: under a best coupling(X1,X2) of Q1 and Q2 , the first part is and the second part is comparison between the locations of X1 and X2 on . Although the coupling method is not very successful in estimating the first part, we prove in this paper that this method can be applied to establish accurate upper bound for the second part. In some cases, the coupling argument can produce exact results. This paper complements the results in Barbour, Brown and Xia (1996)
TL;DR: In this paper, the stability of nonlinear composite stochastic systems with feedback laws was studied and sufficient conditions for the existence of feedback laws which render the equilibrium solution globally asymptotically stable in probability.
Abstract: This paper deals with the stability for a class of nonlinear composite stochastic systems by feedback laws.Firstly,we give sufficient conditions for the existence of feedback laws which render the equilibrium solution of the stochastic system globally asymptotically stable in probability.Secondly,for stochastic systems of the same type,we prove that there exists a linear feedback law which exponentially stabilizes in mean square the closed–loop stochastic system at its equilibrium.
TL;DR: In this article, the kth largest value among the offspring of particles living in the (n-1)st generation of a branching process with immigration was studied. And the results were obtained by combining extreme value theory methods and known results for the behavior of the population size in branching processes with immigration.
Abstract: We study the kth largest value, , among the offspring of particles living in the (n-1)st generation of a branching process with immigration. In particular, stands for the offspring of the most prolific particle in that generation. Limit theorems for are proved. The results are obtained by combining extreme value theory methods and known results for the behavior of the population size in branching processes with immigration
TL;DR: In this article, the exact bound for the convergence of Metropolis chains in a finite state space has been derived based on an interesting observation on the transition probability of the transition probabilities of the chains.
Abstract: In this note, we present a calculation which gives us the exact bound for the convergence of Metropolis chains in a finite state space and therefore improves the existing results which are only for the upper bounds of such convergence (see the references below). Our result is based on an interesting observation on the transition probability of Metropolis chains
TL;DR: In this article, a convex-analytic approach to control discrete-time stochastic processes is proposed. But the convex analytic approach is not suitable for convex spaces and particular Markov models are not considered.
Abstract: Controlled discrete–time stochastic processes axe studied using the convex–analytic approach. Some new properties of strategic measures spaces are established, particular Markov models are considered. The meaningful example is presented.
TL;DR: In this paper, a multistep scheme for computing weak solutions of Lipschitzian quantum stochastic differential equations (QSDE) driven by certain operator-valued Stochastic processes associated with the basic field operators of quantum field theory is introduced and studied.
Abstract: Multistep schemes for computing weak solutions of Lipschitzian quantum stochastic differential equations (QSDE) driven by certain operator-valued stochastic processes associated with the basic field operators of quantum field theory are introduced and studied. This is accomplished within the framework of the Hudson–Parthasarathy formulation of quantum stochastic calculus and subject to matrix element of solution being sufficiently differentiable. Results concerning convergence of explicit schemes of class A in the topology of the locally convex space of solution are presented.Numerical examples are given.
TL;DR: In this paper, the authors study the extension to the case of coefficients depending on the solution X t and show that the representation formula becomes a stochastic integral equation that has to be studied via anticipate calculas.
Abstract: The stochastic variation of constants proved in[2] results to be an interesting tool to study properties of different classes of stochastic differential equations. In particular, we study the extension to the case of coefficients depending on the solution X t. It turns out that the representation formula becomes a stochastic integral equation that has to be studied via anticipate calculas
TL;DR: In this article, the authors studied the existence of the solution to one-dimensional forward-backward stochastic differential equations with neither the smooth condition nor the monotonicity condition for the coefficients.
Abstract: In this paper, we study the existence of the solution to one-dimensional forward–backward stochastic differential equations with neither the smooth condition nor the monotonicity condition for the coefficients. Under the nondegeneracy condition for the forward equation, we prove the existence of the solution to one-dimensional forward–backward stochastic differential equations. And we apply this result to establish the existence of the viscosity solution to a certain one-dimensional quasilinear parabolic partial differential equation
TL;DR: In this paper, a transform-free analysis of the following model is presented, where the state of the system is initially 0 and thereafter increases jumpwise due to compound Poisson shocks.
Abstract: We present a transform–free analysis of the following model. The state of the system is initially 0 and thereafter increases jumpwise due to compound Poisson shocks. Each shock increases the state by a random amount. The system is inspected at random points in time. If the state is above a threshold at an inspection, the system is replaced, otherwise no action is taken. Each replacement instantaneously brings the state back to 0. (Existing models assume either exponential interinspection times or discrete shock magnitudes.) This model can be applied to reliability, inventory, and queueing problems.Interpretations are given throughout to make the results easier to understand and to apply
TL;DR: In this paper, a single product continuous time stochastic inventory model for deteriorating items, driven by a conditional Poisson process, is suggested and a finite dimensional filter for the conditional distribution of Zt is found by observing the history of the inventory level.
Abstract: In this paper, a single product continuous time stochastic inventory model for deteriorating items, driven by a. conditional Poisson process, is suggested. It is assumed the process Zt modulating the jump intensities of the Poisson process is a Markov chain. By observing the history of the inventory level, a finite dimensional filter for the conditional distribution of Zt is found. Further filters are found when the conditional Poisson process is replaced by an integer–valued random measure with predictable compensator depending on a right–constant sample paths process yt
TL;DR: In this paper, the authors studied the regularity of solutions of hyperbolic stochastic partial differential equations by proving that they almost surely belong to the anisotropic Besov-Orlicz space corresponding to the Young function M2(t)= exp (t 2) - 1.
Abstract: In this paper,we firstly study the regularity of solutions of hyperbolic stochastic partial differential equations by proving that they almost surely belong to the anisotropic Besov–Orlicz space corresponding to the Young function M2(t)= exp (t 2) - 1. Secondly, we establish a large deviation principle in this space for the law of the solutions which generalizes the result in Eddahbi [16] dealing with the Hoder topology, weaker than the Besov-Orlicz topology the Strassen's iterated logarithm law for the Brownian sheet obtained in N’zi [29].
TL;DR: In this paper, the authors identify every stable random measure Ψ P → (S, ∥ ∥), Ψ « v, by a vector measure F: P → L α (μ).
Abstract: Let (S, ∥ ∥) be a Banach space of jointly symmetric α-stable random variables and let P be the 6-ring of Borel sets of finite v measure, where v is a regular measure in the real line. In this paper we identify every stable random measure Ψ P → (S, ∥ ∥), Ψ « v, by a vector measure F: P → L α (μ). This leads to a method for identifying spectral domain of a certain class of stable processes including harmonizable processes.
TL;DR: For a multichannel Markovian queue with infinite waiting space, the density function of the busy period can be obtained in series form by using simple induction as mentioned in this paper, which can be used to find easily the moment of the length of the -channel busy period of any arbitrary order in an explicit form.
Abstract: For a multichannel Markovian queue with infinite waiting space the density function of the busy period is obtained in series form by using simple induction.The expression obtained can be used to find easily the moment of the length of the –channel busy period of any arbitrary order in an explicit form.