Showing papers in "Stochastic Analysis and Applications in 2017"
TL;DR: In this article, the existence and asymptotic stability of a mild solution to a class of neutral stochastic integro-differential equations of fractional order involving a mild mild solution was investigated.
Abstract: In this article, we investigate the existence and asymptotic stability in p-th moment of a mild solution to a class of neutral stochastic integro-differential equation of fractional order involving...
29 citations
TL;DR: In this article, the existence and stability of a class of non-instantaneous impulsive fractional-order implicit differential equations with random effects with Ulam-Hyers-Rassias stability was studied.
Abstract: In this article, we study existence and stability of a class of non-instantaneous impulsive fractional-order implicit differential equations with random effects. First, we establish a framework to study impulsive fractional sample path associated with impulsive fractional Lp-problem, and present the relationship between them. We also derive the formula of the solution for inhomogeneous impulsive fractional Lp-problem and sample path. Second, we construct a sequence of Picard functions, which admits us to apply successive approximations method to seek the solution of impulsive fractional sample path. Further, we derive the existence of solutions to impulsive fractional Lp-problem. Third, the concepts of Ulam's type stability are introduced and sufficient conditions to guarantee Ulam–Hyers–Rassias stability are derived. Finally, an example is given to illustrate the theoretical results.
26 citations
TL;DR: In this article, the authors prove the existence of a random attractor for the three-dimensional damped Navier-Stokes equations with additive noise by verifying the pullback flattening property.
Abstract: This article is concerned with the asymptotical behavior of solutions for the three-dimensional damped Navier–Stokes equations with additive noise. Due to the shortage of the existence proof of the existence of random absorbing sets in a more regular phase space, we cannot obtain some kind of compactness of the cocycle associated with the three-dimensional damped Navier–Stokes equations with additive noise by the Sobolev compactness embedding theorem. In this paper, we prove the existence of a random attractor for the three-dimensional damped Navier–Stokes equations with additive noise by verifying the pullback flattening property.
24 citations
TL;DR: In this paper, the stochastic stability in probabilistic SIRS epidemic models with saturated incidence rates and delay has been investigated, and the authors consider stochastically susceptible-infected-removed-susceptible (SIRS) epidemic models.
Abstract: In this article, we consider stochastic susceptible-infected-removed-susceptible (SIRS) epidemic models with saturated incidence rates and delay. We investigate the stochastic stability in probabil...
22 citations
TL;DR: In this paper, the Karhunen-Lo'eve expansion and the Fredholm determinant formula are used to derive an asymptotic Rosenblatt-type distribution of a sequence of integrals of quadratic functions of a Gaussian stationary random field displaying long-range dependence.
Abstract: The Karhunen-Lo`eve expansion and the Fredholm determinant formula are used, to derive
an asymptotic Rosenblatt-type distribution of a sequence of integrals of quadratic functions of
Gaussian stationary random fields on R
d displaying long-range dependence. This distribution
reduces to the usual Rosenblatt distribution when d = 1. Several properties of this new distribution
are obtained. Specifically, its series representation, in terms of independent chi-squared
random variables, is established. Its L´evy-Khintchine representation, and membership to the
Thorin subclass of self-decomposable distributions are obtained as well. The existence and
boundedness of its probability density then follow as a direct consequence.
17 citations
TL;DR: In this paper, the first hitting times of generalized Poisson processes Nf(t) related to Bernstein functions f are studied and the hitting probabilities P{Tαk < ∞} are explicitly obtained and analyzed.
Abstract: In this article, the first hitting times of generalized Poisson processes Nf(t), related to Bernstein functions f are studied. For the space-fractional Poisson processes, Nα(t), t > 0 (corresponding to f = xα), the hitting probabilities P{Tαk < ∞} are explicitly obtained and analyzed. The processes Nf(t) are time-changed Poisson processes N(Hf(t)) with subordinators Hf(t) and here we study and obtain probabilistic features of these extended counting processes. A section of the paper is devoted to processes of the form where are generalized grey Brownian motions. This involves the theory of time-dependent fractional operators of the McBride form. While the time-fractional Poisson process is a renewal process, we prove that the space–time Poisson process is no longer a renewal process.
17 citations
TL;DR: In this paper, the authors obtained a maximal inequality for sub-fractional Brownian motion with Hurst index analogous to the Burkholder-Davis-Gundy inequality for fractional brownian motion derived by Novikov and Valkeila.
Abstract: We obtain a maximal inequality for sub-fractional Brownian motion with Hurst index analogous to the Burkholder–Davis–Gundy inequality for fractional Brownian motion derived by Novikov and Valkeila [Statist. Probab. Lett. 44(1999):47–54] and an integral inequality for Wiener integrals with respect to a sub-fractional Brownian motion with Hurst index .
16 citations
TL;DR: In this paper, the asymptotic behavior of a stochastic Chemostat model with Lotka-Volterra food chain in which the dilution rate was influenced by white noise was studied.
Abstract: This article studies the asymptotic behavior of a stochastic Chemostat model with Lotka–Volterra food chain in which the dilution rate was influenced by white noise. The long-time behavior of the model is studied. Using Lyapunov function and Ito's formula, we show that there is a unique positive solution to the system. Moreover, the sufficient conditions for some population dynamical properties including the boundedness in mean and the stochastically asymptotic stability of the washout equilibrium were obtained. Furthermore, we show how the solutions spiral around the predator-free equilibrium and the positive equilibrium of deterministic system. Besides, the existence of the stationary distribution is proved for the considered model. Numerical simulations are introduced finally to support the obtained results.
15 citations
TL;DR: In this article, the optimality conditions for singular mean-field control are reduced to a reflected Skorohod problem, whose solution is proved to exist uniquely, and sufficient and necessary conditions for the optimal controls and for the Nash equilibrium are obtained.
Abstract: This article studies singular mean field control problems and singular mean field two-players stochastic differential games. Both sufficient and necessary conditions for the optimal controls and for the Nash equilibrium are obtained. Under some assumptions the optimality conditions for singular mean-field control are reduced to a reflected Skorohod problem, whose solution is proved to exist uniquely. Motivations are given as optimal harvesting of stochastic mean-field systems, optimal irreversible investments under uncertainty and mean-field singular investment games. In particular, a simple singular mean-field investment game is studied, where the Nash equilibrium exists but is not unique.
15 citations
TL;DR: In this article, the authors focus on explanation of small-sample effects of normality testing and its robust properties, and embedding these questions into the more general question of testing for sphericity.
Abstract: Stochastic robustness of control systems under random excitation motivates challenging developments in geometric approach to robustness. The assumption of normality is rarely met when analyzing real data and thus the use of classic parametric methods with violated assumptions can result in the inaccurate computation of p-values, effect sizes, and confidence intervals. Therefore, quite naturally, research on robust testing for normality has become a new trend. Robust testing for normality can have counterintuitive behavior, some of the problems have been introduced in Stehlik et al. [Chemometrics and Intelligent Laboratory Systems 130 (2014): 98–108]. Here we concentrate on explanation of small-sample effects of normality testing and its robust properties, and embedding these questions into the more general question of testing for sphericity. We give geometric explanations for the critical tests. It turns out that the tests are robust against changes of the density generating function within the cl...
14 citations
TL;DR: In this paper, a class of stochastic differential equations driven by a fractional Brownian motion with H > 1/2 and a discontinuous coefficient in the diffusion was studied.
Abstract: In this article, we study a class of stochastic differential equations driven by a fractional Brownian motion with H > 1/2 and a discontinuous coefficient in the diffusion. We prove existence and uniqueness for the solution of these equations. This is a first step to define a fractional version of the skew Brownian motion.
TL;DR: In this article, the authors investigate a problem of large deviations for continuous Gaussian Volterra processes, conditioned to follow a fixed trajectory up to a fixed time T > 0, in order to establish the behavior of the process in the near future after T and to give an asymptotic estimate of the exit probability of its bridge.
Abstract: In this article we investigate a problem of large deviations for continuous Gaussian Volterra processes, conditioned to follow a fixed trajectory up to a fixed time T > 0, in order to establish the behavior of the process in the near future after T and to give an asymptotic estimate of the exit probability of its bridge. Some examples are considered.
TL;DR: In this paper, the asymptotic properties of instrumental variable estimators of the drift parameter for stochastic processes driven by mixed fractional Brownian motion were investigated. And they were shown to be robust to a variety of parameters.
Abstract: We investigate the asymptotic properties of instrumental variable estimators of the drift parameter for stochastic processes satisfying linear stochastic differential equations driven by mixed fractional Brownian motion.
TL;DR: In this paper, the multilevel sequential Monte Carlo (MLSMC) method of Beskos et al. is considered in the context of inverse problems, where one discretizes the solution of a partial differential equation.
Abstract: In this article, we consider the multilevel sequential Monte Carlo (MLSMC) method of Beskos et al. (Stoch. Proc. Appl. [to appear]). This is a technique designed to approximate expectations w.r.t. probability laws associated to a discretization. For instance, in the context of inverse problems, where one discretizes the solution of a partial differential equation. The MLSMC approach is especially useful when independent, coupled sampling is not possible. Beskos et al. show that for MLSMC the computational effort to achieve a given error, can be less than independent sampling. In this article we significantly weaken the assumptions of Beskos et al., extending the proofs to non-compact state-spaces. The assumptions are based upon multiplicative drift conditions as in Kontoyiannis and Meyn (Electron. J. Probab. 10 [2005]: 61–123). The assumptions are verified for an example.
TL;DR: In this paper, the authors studied neutral stochastic functional evolution equations driven by jumps modulated by Markovian switching in real separable Hilbert spaces and derived the existence and uniqueness of mild solutions to equations of this class under local non-Lipschitz condition.
Abstract: In this article, we initiate a study on neutral stochastic functional evolution equations driven by jumps modulated by Markovian switching in real separable Hilbert spaces. Our goal here is to derive the existence and uniqueness of mild solutions to equations of this class under local non-Lipschitz condition proposed by Taniguchi [J. Math. Anal. Appl. 340:(2009)197–208] by means of stopping time technique and Banach fixed-point theorem. The results obtained here generalize the main results from Luo and Taniguchi [Stoch. Dyn. 9:(2009)135–152] and Jiang and Shen [Comput. Math. Appl. 61:(2011)1590–1594]. Finally, an example is worked out to illustrate the obtained results.
TL;DR: In this paper, an attempt is made for developing the local lagged adapted generalized method of moments (LLGMM), which is composed of: 1) development of the stochastic model for conti...
Abstract: In this work, an attempt is made for developing the local lagged adapted generalized method of moments (LLGMM). This proposed method is composed of: 1) development of the stochastic model for conti...
TL;DR: In this article, it was shown that this type of bad performance for quadrature of SDEs with infinitely differentiable and bounded coefficients is not a shortcoming of the Euler scheme in particular but can be observed in a worst case sense for every approximation method that is based on finitely many function values of the coefficients of the SDE.
Abstract: In recent work of Hairer, Hutzenthaler and Jentzen, [11], a stochastic differential equation (SDE) with infinitely differentiable andbounded coefficients was constructed such that the Monte Carlo Euler method for approximation of the expected value of the first component of the solution at the final time converges but fails to achieve a mean square error of a polynomial rate. In this article, we show that this type of bad performance for quadrature of SDEs with infinitely differentiable and bounded coefficients is not a shortcoming of the Euler scheme in particular but can be observed in a worst case sense for every approximation method that is based on finitely many function values of the coefficients of the SDE. Even worse we show that for any sequence of Monte Carlo methods based on finitely many sequential evaluations of the coefficients and all their partial derivatives and for every arbitrarily slow convergence speed there exists a sequence of SDEs with infinitely differentiable and bounded ...
TL;DR: In this paper, the authors investigated how to stabilize a given unstable hybrid SDE by feedback controls based on discrete-time state observations, in the sense of H∞, asymptotic and exponential stability in pth moment for all p > 1.
Abstract: Since Mao initiated the study of stabilization of continuous-time hybrid stochastic differential equations (SDEs) by feedback controls based on discrete-time state observations in 2013, many authors have further studied and developed it. However, so far no work on the pth moment stabilization has been reported. This paper is to investigate how to stabilize a given unstable hybrid SDE by feedback controls based on discrete-time state observations, in the sense of H∞, asymptotic and exponential stability in pth moment for all p > 1. The main techniques used are constructions of the Lyapunov functionals and generalizations of inequalities.
TL;DR: In this paper, the mean of the running maximum of the conditional and unconditional Brownian motion of an integrated Gauss-Markov process was derived for the first passage time of the first-passage time.
Abstract: We find explicit formulae for the mean of the running maximum of conditional and unconditional Brownian motion; they are used to obtain the mean, a(t), of the running maximum of an integrated Gauss–Markov process. Then, we deal with the connection between the moments of its first-passage-time and a(t). As explicit examples, we consider integrated Brownian motion and integrated Ornstein–Uhlenbeck process.
TL;DR: In this article, a stochastic logistic model with regime switching modulated by a singular Markov chain involving a small parameter was investigated, where the small parameter is used to reflect rapid rate of regime switching among each state class.
Abstract: Focusing on stochastic dynamics involve continuous states as well as discrete events, this article investigates stochastic logistic model with regime switching modulated by a singular Markov chain involving a small parameter. This Markov chain undergoes weak and strong interactions, where the small parameter is used to reflect rapid rate of regime switching among each state class. Two-time-scale formulation is used to reduce the complexity. We obtain weak convergence of the underlying system so that the limit has much simpler structure. Then we utilize the structure of limit system as a bridge, to invest stochastic permanence of original system driving by a singular Markov chain with a large number of states. Sufficient conditions for stochastic permanence are obtained. A couple of examples and numerical simulations are given to illustrate our results.
TL;DR: In this article, the problem of optimal estimation of the vector parameter θ of the drift term in a sub-fractional Brownian motion was considered and the maximum likelihood estimator as well as Bayesian estimator when the prior distribution is Gaussian was obtained.
Abstract: We consider the problem of optimal estimation of the vector parameter θ of the drift term in a sub-fractional Brownian motion. We obtain the maximum likelihood estimator as well as Bayesian estimator when the prior distribution is Gaussian.
TL;DR: In this paper, the authors introduce an original mean-field particle system, which is always well defined and whose large number particle limit is, in all generality, the distribution of a process conditioned to not hit a given set.
Abstract: The existing literature contains many examples of mean-field particle systems converging to the distribution of a Markov process conditioned to not hit a given set. In many situations, these mean-field particle systems are failable, meaning that they are not well defined after a given random time. Our first aim is to introduce an original mean-field particle system, which is always well defined and whose large number particle limit is, in all generality, the distribution of a process conditioned to not hit a given set. Under natural conditions on the underlying process, we also prove that the convergence holds uniformly in time as the number of particles goes to infinity. As an illustration, we show that our assumptions are satisfied in the case of a piece-wise deterministic Markov process.
TL;DR: In this paper, a class of two-time-scale functional stochastic differential equations, where the phase space of the segment processes is infinite-dimensional, is studied and the ergodicity of the fast-varying component is obtained.
Abstract: This article focuses on a class of two-time-scale functional stochastic differential equations, where the phase space of the segment processes is infinite-dimensional. The systems under consideration have a fast-varying component and a slowly varying one. First, the ergodicity of the fast-varying component is obtained. Then inspired by the Khasminskii’s approach, an averaging principle, in the sense of convergence in the pth moment uniformly in time within a finite time interval, is developed.
TL;DR: In this paper, a portfolio optimization problem with complete memory over a finite time horizon is considered, where the goal is to choose investment and consumption controls such that the total expected discounted utility is maximized.
Abstract: In this article, we consider a portfolio optimization problem of the Merton’s type with complete memory over a finite time horizon. The problem is formulated as a stochastic control problem on a finite time horizon and the state evolves according to a process governed by a stochastic process with memory. The goal is to choose investment and consumption controls such that the total expected discounted utility is maximized. Under certain conditions, we derive the explicit solutions for the associated Hamilton–Jacobi–Bellman (HJB) equations in a finite-dimensional space for exponential, logarithmic, and power utility functions. For those utility functions, verification results are established to ensure that the solutions are equal to the value functions, and the optimal controls are also derived.
TL;DR: It is proved that when the colored noise tends to the white noise, the random attractor of the perturbed system with colored noise converges to that of the original SPDEs by invoking some continuity results on attractors in random dynamical systems.
Abstract: We use the method of smooth approximation to examine the random attractor for two classes of stochastic partial differential equations (SPDEs). Roughly speaking, we perturb the SPDEs by a Wong-Zakai scheme using smooth colored noise approximation rather than the usual polygonal approximation. After establishing the existence of the random attractor of the perturbed system, we prove that when the colored noise tends to the white noise, the random attractor of the perturbed system with colored noise converges to that of the original SPDEs by invoking some continuity results on attractors in random dynamical systems.
TL;DR: In this paper, a novel approach of calculating an insurance premium based on g-integrals and interval-valued integrals is introduced, and the characterization theorem for the gintegral-based premium principle is proven, and relations with some well-known premium principles are discussed.
Abstract: A novel approach of calculating an insurance premium based on g-integrals and interval-valued integrals is introduced. The characterization theorem for the g-integral-based premium principle is proven, and the relations with some well-known premium principles are discussed. The main properties of the interval-valued premium principle based on the g-integral are presented and some illustrative examples are given.
TL;DR: In this article, almost automorphic solutions for semilinear stochastic differential equations driven by Levy noise were studied and the existence and uniqueness of bounded solutions were established by using the Banach fixed point theorem and the exponential dichotomy property.
Abstract: In this article, we study almost automorphic solutions for semilinear stochastic differential equations driven by Levy noise. We establish the existence and uniqueness of bounded solutions by using the Banach fixed point theorem, the exponential dichotomy property and stochastic analysis techniques. Furthermore, this unique bounded solution is almost automorphic in distribution under slightly stronger conditions. We also give two examples to illustrate our results.
TL;DR: In this paper, the authors characterize continuous linear operators from to via their 2D-Fock transforms and show that there exists a one-to-one correspondence between continuous linear operator from to and functions on Γ × Γ that satisfy some type of growth condition, where Γ designates the finite power set.
Abstract: In this article, we aim at characterizing operators acting on functionals of discrete-time normal martingales. Let be a discrete-time normal martingale that has the chaotic representation property. We first introduce a transform, called 2D-Fock transform, for operators from the testing functional space to the generalized functional space of M. Then we characterize continuous linear operators from to via their 2D-Fock transforms. Our characterization theorems show that there exists a one-to-one correspondence between continuous linear operators from to and functions on Γ × Γ that only satisfy some type of growth condition, where Γ designates the finite power set of . Finally, we give some applications of our characterization theorems.
TL;DR: In this article, the authors study a flexible and simplistic model of interest rate, w.r.t. negative or close to zero interest rates, in the context of the recent global financial crisis.
Abstract: The recent global financial crisis caused implementation of negative or close to zero interest rates. This situation implies a necessity to study a flexible and simplistic model of interest rate, w...
TL;DR: In this article, the problem of maximizing expected utility from the terminal wealth with proportional transaction costs and random endowment is studied in the context of the existence of consistent utility maximization.
Abstract: In this article, we study the problem of maximizing expected utility from the terminal wealth with proportional transaction costs and random endowment. In the context of the existence of consistent...