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Showing papers in "Stochastic Analysis and Applications in 2009"


Journal ArticleDOI
TL;DR: In this paper, the modulus of continuity of a stochastic process is defined as a random element for any fixed mesh size, and the convergence rate of Euler-Maruyama schemes with uniformly bounded coefficients is analyzed.
Abstract: The modulus of continuity of a stochastic process is a random element for any fixed mesh size. We provide upper bounds for the moments of the modulus of continuity of Ito processes with possibly unbounded coefficients, starting from the special case of Brownian motion. References to known results for the case of Brownian motion and Ito processes with uniformly bounded coefficients are included. As an application, we obtain the rate of strong convergence of Euler–Maruyama schemes for the approximation of stochastic delay differential equations satisfying a Lipschitz condition in supremum norm.

87 citations


Journal ArticleDOI
TL;DR: In this paper, Lakshmikantham and Ciric proved coupled random coincidence and coupled random fixed point theorems for nonlinear contractions in partially ordered metric spaces with mixed g-monotone properties.
Abstract: Let (X, ≤) be a partially ordered set and suppose there is a metric d on X such that (X, d) is a complete separable metric space and (Ω, Σ) be a measurable space. In this article a pair of random mappings F: Ω × (X × X) → X and g: Ω × X → X, where F has a mixed g-monotone property on X, and F and g satisfy the non-linear contractive condition (5) below, are introduced and investigated. Two coupled random coincidence and coupled random fixed point theorems are proved. These results are random versions and extensions of recent results of Lakshmikantham and Ciric [V. Lakshmikantham and Lj. Ciric, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal.—Theor. 70(12) (2009): 4341–4349] and include several recent developments.

64 citations


Journal ArticleDOI
TL;DR: In this paper, the existence and asymptotical stability of mild solutions to stochastic neutral partial differential equations with infinite delays was studied. But the stability of these solutions was not investigated.
Abstract: In this article, we study the existence and the asymptotical stability in mean square of mild solutions to stochastic neutral partial differential equations with infinite delays where t − τ(t), t − δ(t), t − ρ(t) → ∞ with delays τ(t), δ(t), ρ(t) → ∞ as t → ∞. Our method for investigating the stability of solutions is based on the fixed point theorem.

53 citations


Journal ArticleDOI
TL;DR: In this paper, the authors stochastically perturb the classical non-autonomous Lotka-Volterra model into the stochastic differential equation, and then establish new asymptotic properties for the moments and sample paths of the solution.
Abstract: In this article, we stochastically perturb the classical non-autonomous Lotka–Volterra model into the stochastic differential equation Different from most of existing articles, for example, [3, 20] the system parameters in this article are time-dependent. We will give a sufficient condition under which the stochastic differential equation will have a unique global positive solution. We will then establish some new asymptotic properties for the moments as well as for the sample paths of the solution. In particular, we will discuss two fundamental problems in population systems, namely ultimate boundedness and extinction.

44 citations


Journal ArticleDOI
TL;DR: In this paper, a Donsker type approximation theorem for the Rosenblatt process is proved and a binary market model driven by this process is constructed, which admits arbitrage opportunities.
Abstract: In this article, we prove a Donsker type approximation theorem for the Rosenblatt process, which is a selfsimilar stochastic process exhibiting long range dependence. We use this result to construct a binary market model driven by this process and we show that the model admits arbitrage opportunities. Finally we present some numerical simulations to illustrate the method of approximation.

44 citations


Journal ArticleDOI
TL;DR: In this paper, the strong convergence of the Euler-Maruyama method and stochastic theta method with jumps was investigated under a local Lipschitz condition and a linear growth condition.
Abstract: In this article, we investigate the strong convergence of the Euler–Maruyama method and stochastic theta method for stochastic differential delay equations with jumps. Under a global Lipschitz condition, we not only prove the strong convergence, but also obtain the rate of convergence. We show strong convergence under a local Lipschitz condition and a linear growth condition. Moreover, it is the first time that we obtain the rate of the strong convergence under a local Lipschitz condition and a linear growth condition, i.e., if the local Lipschitz constants for balls of radius R are supposed to grow not faster than log R.

38 citations


Journal ArticleDOI
TL;DR: In this paper, a numerical method using stochastic approximation approach for an optimal trading (buy and sell) strategy is proposed, where the underlying asset price is governed by a mean re-reverting process.
Abstract: This article is concerned with a numerical method using stochastic approximation approach for an optimal trading (buy and sell) strategy. The underlying asset price is governed by a mean-reverting stochastic process. The objective is to buy and sell the asset so as to maximize an overall expected return. One of the advantages of our approach is that the underlying asset is model free. Only mean reversion is required. Slippage cost is imposed on each transaction. Convergence of the algorithms is provided. Numerical examples are reported to demonstrate the results.

37 citations


Journal ArticleDOI
TL;DR: This research considers a mathematical model of tumor-virus-immune system dynamics and incorporates random noise inherent in any physiological process by extending the deterministic model into a stochastic one.
Abstract: Ongoing research on cancer has indicated that viruses which specifically infect and destroy tumor cells could be used as a therapeutic agent to arrest tumors Some of these oncolytic viruses had already been tested clinically with success However, the interaction dynamics between the tumor host, the invading virus and the immune system response is highly nonlinear and complex and, hence, a proper understanding of such dynamics requires involvement of mathematical models In the present research, we consider a mathematical model of tumor-virus-immune system dynamics We analyze the basic deterministic model to find out the importance of different host, viral and immune system parameters in controlling the system dynamics Next we incorporate random noise inherent in any physiological process by extending the deterministic model into a stochastic one The resulting stochastic model is studied using the mean square stability approach and criteria for stochastic stability is derived in terms of impo

33 citations


Journal ArticleDOI
TL;DR: In this article, a stochastic partial differential equation model under a fast random dynamical boundary condition is investigated, where the noises in the model and in the boundary condition are both additive.
Abstract: In order to understand the impact of random influences at physical boundary on the evolution of multiscale systems, a stochastic partial differential equation model under a fast random dynamical boundary condition is investigated The noises in the model and in the boundary condition are both additive An effective equation is derived and justified by reducing the random dynamical boundary condition to a simpler one The effective system is still a stochastic partial differential equation Furthermore, the quantitative comparison between the solution of the original stochastic system and the effective solution is provided by establishing normal deviations and large deviations principles Namely, the normal deviations are asymptotically characterized, while the rate and speed of the large deviations are estimated

32 citations


Journal ArticleDOI
TL;DR: In this article, the authors present a rigorous mathematical derivation of a macroscopic model of aggregation, scaling up from a microscopic description of a family of individuals subject to aggregation/repulsion, described by a system of Ito type stochastic differential equations.
Abstract: In this article, we present a rigorous mathematical derivation of a macroscopic model of aggregation, scaling up from a microscopic description of a family of individuals subject to aggregation/repulsion, described by a system of Ito type stochastic differential equations. We analyze the asymptotics of the system for both a large number of particles on a bounded time interval, and its long time behavior, for a fixed number of particles. As far as this second part is concerned, we show that a suitable localizing potential is required, in order that the system may admit a nontrivial invariant distribution.

32 citations


Journal ArticleDOI
TL;DR: In this article, a wide class of non-Volterra quadratic stochastic operators using N. Ganikhadjaev's construction of quadrastic stochastically operators is described.
Abstract: In this article, we describe a wide class of non-Volterra quadratic stochastic operators using N. Ganikhadjaev's construction of quadratic stochastic operators. By the construction these operators depend on a probability measure μ being defined on the set of all configurations which are given on a graph G. We show that if μ is the product of probability measures being defined on each maximal connected subgraphs of G then corresponding non-Volterra operator can be reduced to m number (where m is the number of maximal connected subgraphs of G) of Volterra operators defined on the maximal connected subgraphs. Our result allows to study a wide class of non-Volterra operators in the framework of the well known theory of Volterra quadratic stochastic operators.

Journal ArticleDOI
TL;DR: In this paper, the authors examined a finite-dimensional linear inverse problem where the measurements are disturbed by an additive normal noise, and the problem is solved both in the frequentist and in the Bayesian frameworks.
Abstract: In this work, we examine a finite-dimensional linear inverse problem where the measurements are disturbed by an additive normal noise. The problem is solved both in the frequentist and in the Bayesian frameworks. Convergence of the used methods when the noise tends to zero is studied in the Ky Fan metric. The obtained convergence rate results and parameter choice rules are of a similar structure for both approaches.

Journal ArticleDOI
TL;DR: In this paper, a procedure based on fuzzy set theory is introduced that allows evaluating and comparing recruitment strategies, resulting in the determination of a most preferable strategy, under stochastic assumptions for attainability after one step.
Abstract: One aspect of Manpower Planning is the investigation of wastage and internal transitions for homogeneous groups of employees in a Manpower system. In the relevant literature, the attainability of a desired stock vector was studied under control by recruitment for time-discrete Markov models. These approaches allow choosing a proper recruitment strategy, resulting in an attainable vector most similar to a desired one. In this article, this problem will be discussed under stochastic assumptions for attainability after one step. Based on fuzzy set theory, a procedure is introduced that allows evaluating and comparing recruitment strategies, resulting in the determination of a most preferable strategy.

Journal ArticleDOI
TL;DR: In this paper, a batch arrival queueing system with a single vacation between two successive busy periods and with exhaustive service is considered, where the departure process h(t) is studied first on a one vacation cycle and the explicit representation for the generating function of Laplace transform of the probability function of h( t) is derived.
Abstract: A batch arrival queueing system with a single vacation between two successive busy periods and with exhaustive service is considered. The departure process h(t) is studied first on a single vacation cycle. The approach based on renewal theory is applied to obtain results in the general case. In particular, the explicit representation for the generating function of Laplace transform of the probability function of h(t) is derived. All formulae are written in terms of input parameters of the system and factors of a certain canonical factorization of Wiener–Hopf type. A numerical approach to results is discussed as well.

Journal ArticleDOI
TL;DR: In this article, the authors study a large financial market where the discounted asset prices are modeled by martingale random fields and study the minimal variance hedging problem both in the case of full and partial information.
Abstract: We study a large financial market where the discounted asset prices are modeled by martingale random fields This approach allows the treatment of both the cases of a market with a countable amount of assets and of a market with a continuum amount We discuss conditions for these markets to be complete and we study the minimal variance hedging problem both in the case of full and partial information An explicit representation of the minimal variance hedging portfolio is suggested Techniques of stochastic differentiation are applied to achieve the main results Examples of large market models with a countable number of assets are considered according to the literature and an example of market model with a continuum of assets is taken from the bond market

Journal ArticleDOI
TL;DR: In this paper, the authors presented the probability generating function of the stationary queue length and sojourn time for the GI/Geo/1 queue with disasters, and converted the results into the G/M/1 and G/G/1 queues.
Abstract: The occurrence of disasters to a queueing system causes all customers to be removed if any are present. Although there has been much research on continuous-time queues with disasters, the discrete-time Geo/Geo/1 queue with disasters has appeared in the literature only recently. We extend this Geo/Geo/1 queue to the GI/Geo/1 queue. We present the probability generating function of the stationary queue length and sojourn time for the GI/Geo/1 queue. In addition, we convert our results into the Geo/Geo/1 queue and the GI/M/1 queue.

Journal ArticleDOI
TL;DR: In this paper, a central limit theorem for the finite-dimensional laws of the quadratic variations process of certain fractional Brownian sheets is proved for the Malliavin calculus.
Abstract: In this article, we state and prove a central limit theorem for the finite-dimensional laws of the quadratic variations process of certain fractional Brownian sheets. The main tool of this article is a method developed by Nourdin and Nualart in [18] based on the Malliavin calculus.

Journal ArticleDOI
TL;DR: In this paper, a probabilistic solution of a non degenerate Poisson type equation with Neumann boundary condition in a bounded domain of the Euclidean space is given.
Abstract: In this work we extend Brosamler's formula (see “A probabalistic solution to the Neumann problem,” Math. Scand. 1976, 38:137–147) and give a probabilistic solution of a non degenerate Poisson type equation with Neumann boundary condition in a bounded domain of the Euclidean space.

Journal ArticleDOI
TL;DR: In this article, a design algorithm for a memoryless state feedback controller which guarantees that the closed-loop dynamics will be regular, impulse-free, and robust stable is proposed in terms of the solutions to linear matrix inequalities (LMIs).
Abstract: This article deals with the class of continuous-time singular uncertain linear systems with time-varying delay in the state vector. The uncertainties we are considering are of norm bounded type. Delay-dependent sufficient conditions on robust stability and robust stabilizability are developed. A design algorithm for a memoryless state feedback controller which guarantees that the closed-loop dynamics will be regular, impulse-free, and robust stable is proposed in terms of the solutions to linear matrix inequalities (LMIs).

Journal ArticleDOI
TL;DR: In this article, the point-wise ergodicity of α-stable systems with sufficiently small interaction in a large subspace of R Z d has been proved for finite and infinite dimensions.
Abstract: Some finite and infinite dimensional perturbed α-stable dynamics are constructed and studied in this article. We prove that the finite dimensional system is strongly mixing, while in the infinite dimensional case that the functional coercive inequalities are not available, we develop and apply a technique to prove the point-wise ergodicity for systems with sufficiently small interaction in a large subspace of Ω = R Z d .

Journal ArticleDOI
TL;DR: In this paper, a white noise calculus for fractional Brownian motion with values in a separable Hilbert space is considered, where the covariance operator Q is a kernel operator (Q-fractional brownian motion).
Abstract: In this article, we consider a white noise calculus for fractional Brownian motion with values in a separable Hilbert space, whereby the covariance operator Q is a kernel operator (Q-fractional Brownian motion). We introduce a fractional exponential, Q-fractional test function space , and a corresponding distribution space and the Q-fractional version of the Wick product, so that we can define a stochastic integral with respect to a fractional white noise process. Furthermore, a Q-fractional version of the Girsanov formula and a Q-fractional version of the Clark–Haussmann–Ocone theorem are proved. As applications, the solution for a stochastic evolution equation driven by fractional Brownian motion and an infinite-dimensional fractional Black–Scholes market are discussed.

Journal ArticleDOI
TL;DR: In this article, the properties of multifractal products of the exponential of stationary diffusion processes defined by stochastic differential equations with linear drift and certain form of the diffusion coefficient corresponding to a variety of marginal distributions are investigated.
Abstract: We investigate the properties of multifractal products of the exponential of stationary diffusion processes defined by stochastic differential equations with linear drift and certain form of the diffusion coefficient corresponding to a variety of marginal distributions. The conditions on the mean, variance and covariance functions of these processes are interpreted in terms of the moment generating functions. We provide three illustrative examples of normal, gamma and beta distributions. We establish the corresponding lognormal, log-gamma and log-beta scenarios for the limiting processes, including their Renyi functions and dependence structure.

Journal ArticleDOI
TL;DR: In this article, the authors explore the Erlang series approach for the first-time passage problem for jump-diffusions with polynomial state-dependent coefficients and identify cases in which the expansion is finite and the recurrence is of second order.
Abstract: We explore the Erlang series approach for the first-time passage problem for a particular class of jump-diffusions with polynomial state-dependent coefficients. This approach may be viewed as a discrete analog of the Laplace transform, which replaces the differential equations with polynomial coefficients satisfied by this function by algebraic recurrences. We identify cases in which the expansion is finite and in which the recurrence is of second order, and thus more easily solved.

Journal ArticleDOI
TL;DR: In this paper, the authors developed and applied a numerical scheme for pricing options in the stochastic volatility model proposed by Barndorff-Nielsen and Shephard.
Abstract: We develop and apply a numerical scheme for pricing options in the stochastic volatility model proposed by Barndorff–Nielsen and Shephard. This non-Gaussian Ornstein–Uhlenbeck type of volatility model gives rise to an incomplete market, and we consider the option prices under the minimal entropy martingale measure. To numerically price options with respect to this risk neutral measure, one needs to consider a Black and Scholes type of partial differential equation, with an integro-term arising from the volatility process. We suggest finite difference schemes to solve this parabolic integro-partial differential equation, and derive appropriate boundary conditions for the finite difference method. As an application of our algorithm, we consider price deviations from the Black and Scholes formula for call options, and the implications of the stochastic volatility on the shape of the volatility smile.

Journal ArticleDOI
TL;DR: In this article, the authors dealt with expected average reward and sensitive discount criteria for a general class of Markov diffusion processes and gave conditions under which average reward optimality and strong−− 1-discount optimality are equivalent.
Abstract: This article deals with expected average reward (aka ergodic reward) and sensitive discount criteria for a general class of Markov diffusion processes We give conditions under which: (1) average reward optimality and strong − 1-discount optimality are equivalent; (2) strong 0-discount optimality implies bias optimality; and (3) bias optimality implies 0-discount optimality Moreover, under additional hypotheses, (4) bias optimality also implies strong 0-discount optimality Thus, under previous results that guarantee average and bias optimality, we ensure that strong − 1-discount and strong 0-discount optimality hold

Journal ArticleDOI
TL;DR: In this paper, a Geo/G/1/∞ queueing system under multiple vacations and setup-closedown times was investigated. But the authors focused on the service length of the system and the main performance measures.
Abstract: This article concerns a Geo/G/1/∞ queueing system under multiple vacations and setup-closedown times. Specifically, the operation of the system is as follows. After each departure leaving an empty system, the server is deactivated during a closedown time. At the end of each closedown time, if at least a customer is present in the system, the server begins the service of the customers (is reactivated) without setup; however, if the system is completely empty, the server takes a vacation. At the end of each vacation, if there is at least a customer in the system, the server requires a startup time (is reactivated) before beginning the service of the customers; nevertheless, if there are not customers waiting in the system, the server takes another vacation. By applying the supplementary variable technique, the joint generating function of the server state and the system length together with the main performance measures are derived. We also study the length of the different busy periods of the serv...

Journal ArticleDOI
TL;DR: In this paper, a coupled system of the two-dimensional Navier-Stokes equations and the salinity transport equation with spatially correlated white noise on the boundary as well as in fluid is investigated.
Abstract: A coupled system of the two-dimensional Navier–Stokes equations and the salinity transport equation with spatially correlated white noise on the boundary as well as in fluid is investigated. The noise affects the system through a dynamical boundary condition. This system may be considered as a model for gravity currents in oceanic fluids. The noise is due to uncertainty in salinity flux on fluid boundary. After transforming this system into a random dynamical system, we first obtain asymptotic estimates on system evolution, and then show that the long time dynamics is captured by a random attractor.

Journal ArticleDOI
TL;DR: In this article, the problem of p-moment stability of stochastic differential delay equations with impulsive jump and Markovian switching was studied. And the authors obtained new criteria for ensuring pmoments stability of trivial solution of a class of impulsive stochastically differential delay problems with Markovians switching.
Abstract: This article is concerned with the problem of p-moment stability of stochastic differential delay equations with impulsive jump and Markovian switching. In this model, the features of stochastic systems, delay systems, impulsive systems, and Markovian switching are all taken into account, which is scarce in the literature. Based on Lyapunov–Krasovskii functional method and stochastic analysis theory, we obtain new criteria ensuring p-moment stability of trivial solution of a class of impulsive stochastic differential delay equations with Markovian switching.

Journal ArticleDOI
TL;DR: In this article, the authors study the parabolic Volterra equation on a separable Hilbert space and derive optimal conditions for the existence of a unique mild solution and the Holderianity of its trajectories.
Abstract: Aim of this article is to study the parabolic Volterra equation on a separable Hilbert space. Throughout this work the operator −A is assumed to be a differential operator like the Laplacian, the elasticity operator, or the Stokes operator. The random disturbance Q 1/2 is modeled to be a system independent vector valued fractional Brownian motion with Hurst parameter ∊ (0, 1). We derive optimal conditions for the existence of a unique mild solution and the Holderianity of its trajectories. For this purpose we do the analysis on stochastic integrals of the form where the integrand R is a deterministic, operator valued function.

Journal ArticleDOI
TL;DR: In this article, a single-server queue with renewal input and Markovian service process where server serves customers in batches according to a general bulk service rule is considered, and queue length distributions at pre-arrival and arbitrary epochs have been obtained along with some important performance measures such as probability of blocking, mean queue lengths and mean waiting times.
Abstract: We consider a single-server queue with renewal input and Markovian service process where server serves customers in batches according to a general bulk service rule Queue length distributions at pre-arrival and arbitrary epochs have been obtained along with some important performance measures such as probability of blocking, mean queue lengths and mean waiting times The analysis has been carried out assuming finite-buffer space for the arriving customers The model has potential applications in areas such as computer networks, telecommunication systems and manufacturing systems