Showing papers in "Stochastic Analysis and Applications in 2016"
TL;DR: In this article, the existence and uniqueness of the square-mean pseudo almost automorphic mild solutions for a class of stochastic evolution equations driven by G-Brownian motion (G-SSEs) were established by means of the fixed point theorem.
Abstract: In this article, we prove the existence and uniqueness of the square-mean pseudo almost automorphic mild solutions for a class of stochastic evolution equations driven by G-Brownian motion (G-SSEs). Our results are established by means of the fixed point theorem. An example is given to illustrate the theory.
26 citations
TL;DR: In this paper, the controllability results of neutral stochastic delay partial functional integro-differential equations perturbed by fractional Brownian motion are investigated and sufficient conditions are established using the theory of resolvent operators developed by Grimmer.
Abstract: This article focuses on controllability results of neutral stochastic delay partial functional integro-differential equations perturbed by fractional Brownian motion. Sufficient conditions are established using the theory of resolvent operators developed by Grimmer [Resolvent operators for integral equations in Banach spaces, Trans. Amer. Math. Soc., 273(1982):333–349] combined with a fixed point approach for achieving the required result. An example is provided to illustrate the theory.
25 citations
TL;DR: In this article, the authors studied the two-dimensional stochastic Boussinesq system with zero dissipation and multiplicative noise, and showed the existence of a martingale solution by a priori estimates using stocho-calculus, and applications of Prokhorov's, Skorokhod's, and Martingale representation theorems.
Abstract: We study the two-dimensional stochastic Boussinesq system with zero dissipation and multiplicative noise. We show the existence of a martingale solution by a priori estimates using stochastic calculus, and applications of Prokhorov's, Skorokhod's, and martingale representation theorems. Due to the lack of dissipation, the proof requires higher regularity estimates, taking advantage of the structure of the nonlinear term. Moreover, we obtain the existence of the pressure term via an application of de Rham's theorem for processes.
24 citations
TL;DR: Deterministic and stochastic models of cancer-virus dynamics are constructed and viruses with high infection rates and optimal cytotoxicity are suggested to be effective for cancer treatment.
Abstract: Cancer virotherapy is studied in mathematical modeling to improve tumor elimination. Since various oncolytic viruses are used for cancer therapy and virus selection is an important research problem, we, therefore, constructed deterministic and stochastic models of cancer-virus dynamics. We investigated virus characteristic parameter sensitivities using a reproduction ratio. Locally and globally asymptotically stable equilibrium points that are respectively related to therapy failure/partial success and therapy failure were determined. A stochastic system was derived from the deterministic model. Tumor extinction probabilities depending on changing parameter values were investigated. Results suggest that viruses with high infection rates and optimal cytotoxicity are effective for cancer treatment.
22 citations
TL;DR: In this paper, the authors studied infinite horizon discounted-cost and ergodic-cost risk-sensitive zero-sum stochastic games for controlled continuous time Markov chains on a countable state space.
Abstract: We study infinite horizon discounted-cost and ergodic-cost risk-sensitive zero-sum stochastic games for controlled continuous time Markov chains on a countable state space. For the discounted-cost game, we prove the existence of value and saddle-point equilibrium in the class of Markov strategies under nominal conditions. For the ergodic-cost game, we prove the existence of values and saddle point equilibrium by studying the corresponding Hamilton-Jacobi-Isaacs equation under a certain Lyapunov condition.
20 citations
TL;DR: In this paper, a stochastic representation for a vector random field that is stationary, isotropic, and mean square continuous on a sphere or unit circle is derived, where the sequence of ultraspherical or Gegenbauer's polynomials look like a mimic of the series representation of the covariance matrix function.
Abstract: A stochastic representation is derived for a vector random field that is stationary, isotropic, and mean square continuous on a sphere or unit circle. The established stochastic representation is an infinite series involving the sequence of ultraspherical or Gegenbauer's polynomials, looks like a mimic of the series representation of the covariance matrix function of the isotropic vector random field, but differs from the spectral representation in terms of the ordinary spherical harmonics. It is also shown in this paper that some isotropic and continuous covariance matrix functions on the real line or , if they are compactly supported, can be adopted as covariance matrix functions on the unit circle or .
19 citations
TL;DR: In this paper, a deviation inequality for Gaussian approximation of dynamical systems with small random perturbations under Holder norms is presented and the moderate deviation principle and the central limit theorem for the dynamical system by the deviation inequality.
Abstract: In this article, we consider asymptotic behaviors for functionals of dynamical systems with small random perturbations. First, we present a deviation inequality for Gaussian approximation of dynamical systems with small random perturbations under Holder norms and establish the moderate deviation principle and the central limit theorem for the dynamical systems by the deviation inequality. Then, applying these results to forward-backward stochastic differential equations and diffusions in small time intervals, combining the delta method in large deviations, we give a moderate deviation principle for solutions of forward-backward stochastic differential equations with small random perturbations, and obtain the central limit theorem, the moderate deviation principle and the iterated logarithm law for functionals of diffusions in small time intervals.
17 citations
TL;DR: In this article, a class of stochastic reaction-diffusion equations with additive noise is considered, where the solution of a suitable system of ordinary differential equation only describing the reactions, but due to nonlinear interaction of large diffusion and fluctuations in the limit new effective reaction terms appear.
Abstract: We consider a class of a stochastic reaction-diffusion equations with additive noise. In the limit of fast diffusion, one can approximate solutions of the stochastic reaction–diffusion equations by the solution of a suitable system of ordinary differential equation only describing the reactions, but due to nonlinear interaction of large diffusion and fluctuations in the limit new effective reaction terms appear. We focus on systems with polynomial nonlinearities and illustrate the result by applying it to a predator-prey system and a cubic auto-catalytic reaction between two chemicals.
17 citations
TL;DR: In this paper, the authors studied the approximate controllability of an impulsive semilinear stochastic system with delay in state in Hilbert spaces, and obtained the sufficient conditions for such a system to be controllable.
Abstract: Many practical systems in physical and biological sciences have impulsive dynamical behaviors during the evolution process that can be modeled by impulsive differential equations. This article studies the approximate controllability of impulsive semilinear stochastic system with delay in state in Hilbert spaces. Assuming the conditions for the approximate controllability of the corresponding deterministic linear system, we obtain the sufficient conditions for the approximate controllability of the impulsive semilinear stochastic system with delay in state. The results are obtained by using Banach fixed point theorem. Finally, two examples are given to illustrate the developed theory.
14 citations
TL;DR: In this paper, the existence and uniqueness of mild solution for a system of semilinear impulsive differential equations with infinite fractional Brownian motions is proved, and the relationship between mild and weak solutions and the exponential stability of mild solutions are investigated.
Abstract: Some results on the existence and uniqueness of mild solution for a system of semilinear impulsive differential equations with infinite fractional Brownian motions are proved. The approach is based on Perov's fixed point theorem and a new version of Schaefer's fixed point theorem in generalized Banach spaces. The relationship between mild and weak solutions and the exponential stability of mild solutions are investigated as well. The abstract theory is illustrated with an example.
14 citations
TL;DR: In this paper, the optimal consumption and investment policy of an agent who has a quadratic felicity function and faces a subsistence consumption constraint is analyzed. And the agent's optimal investment in the risky asset increases linearly for low wealth levels.
Abstract: In this article, we analyze the optimal consumption and investment policy of an agent who has a quadratic felicity function and faces a subsistence consumption constraint. The agent's optimal investment in the risky asset increases linearly for low wealth levels. Risk taking continues to increase at a decreasing rate for wealth levels higher than subsistence wealth until it hits a maximum at a certain wealth level, and declines for wealth levels above this threshold. Further, the agent has a bliss level of consumption, since if an agent consumes more than this level she will suffer utility loss. Eventually her risk taking becomes zero at a wealth level which supports her bliss consumption.
TL;DR: In this paper, a new stochastic model for the point kinetics equations with I-delayed neutron precursor groups is presented, and the mean and standard deviation of neutron and precursor populations with step, ramp, and sinusoidal reactivities are computed.
Abstract: A new stochastic model for the point kinetics equations with I-delayed neutron precursor groups is presented. In this stochastic model, the point kinetics equations are separated into three terms: prompt neutrons, delayed neutrons and external neutrons source. The matrix form of the efficient stochastic model is solved by a semi-analytical method. The semi-analytical method is based on the exponential function of the coefficient matrix. The eigenvalues of the coefficient matrix and Gaussian elimination are used to calculate this exponential function. The mean and standard deviation of neutron and precursor populations of the efficient stochastic model with step, ramp, and sinusoidal reactivities are computed. The results of the efficient stochastic model are compared with the results of Allen's stochastic model for the point kinetics equations. This comparison confirms that the efficient stochastic model is an accurate model compared with the deterministic point kinetics equations. This stochastic...
TL;DR: In this paper, the authors established the existence of a random attractor for the three-dimensional planetary geostrophic equations of large-scale ocean circulation with small multiplicative noise by verifying the pullback flattening property.
Abstract: The aim of this article is to study the asymptotical behavior, in terms of upper semi-continuous property of attractor, for small multiplicative noise of the three-dimensional planetary geostrophic equations of large-scale ocean circulation. In this article, we establish the existence of a random attractor for the three-dimensional planetary geostrophic equations of large-scale ocean circulation with small multiplicative noise by verifying the pullback flattening property and prove that the random attractor of the three-dimensional planetary geostrophic equations of large-scale ocean circulation with small multiplicative noise converges to the global attractor of the unperturbed three-dimensional planetary geostrophic equations of large-scale ocean circulation when the parameter of the perturbation tends to zero.
TL;DR: In this article, the authors combine stochastic control methods, white noise analysis, and Hida-Malliavin calculus applied to the Donsker delta functional to obtain explicit representations of semimartingale decompositions under enlargement of filtrations.
Abstract: We combine stochastic control methods, white noise analysis, and Hida–Malliavin calculus applied to the Donsker delta functional to obtain explicit representations of semimartingale decompositions under enlargement of filtrations. Some of the expressions are more explicit than previously known. The results are illustrated by examples.
TL;DR: In this paper, Liu et al. studied stationary solutions of stochastic linear retarded functional differential equations with the emphasis on delays which appear in those terms including spatial partial derivatives, and presented conditions on the delay systems to obtain a unique stationary solution by combining spectrum analysis of unbounded operators and stochiastic calculus.
Abstract: This article continues the study of Liu [Statist. Probab. Lett. 78(2008): 1775–1783; Stoch. Anal. Appl. 29(2011): 799–823] for stationary solutions of stochastic linear retarded functional differential equations with the emphasis on delays which appear in those terms including spatial partial derivatives. As a consequence, the associated stochastic equations have unbounded operators acting on the point or distributed delayed terms, while the operator acting on the instantaneous term generates a strongly continuous semigroup. We present conditions on the delay systems to obtain a unique stationary solution by combining spectrum analysis of unbounded operators and stochastic calculus. A few instructive cases are analyzed in detail to clarify the underlying complexity in the study of systems with unbounded delayed operators.
TL;DR: In this paper, the authors present an extension of the forward-reverse representation introduced by Bayer and Schoenmakers (Annals of Applied Probability, 24(5):1994-2032, 2014) to the context of stochastic reaction networks (SRNs).
Abstract: In this work, we present an extension of the forward–reverse representation introduced by Bayer and Schoenmakers (Annals of Applied Probability, 24(5):1994–2032, 2014) to the context of stochastic reaction networks (SRNs). We apply this stochastic representation to the computation of efficient approximations of expected values of functionals of SRN bridges, that is, SRNs conditional on their values in the extremes of given time intervals. We then employ this SRN bridge-generation technique to the statistical inference problem of approximating reaction propensities based on discretely observed data. To this end, we introduce a two-phase iterative inference method in which, during phase I, we solve a set of deterministic optimization problems where the SRNs are replaced by their reaction-rate ordinary differential equations approximation; then, during phase II, we apply the Monte Carlo version of the expectation-maximization algorithm to the phase I output. By selecting a set of overdispersed seeds ...
TL;DR: In this article, the covariance matrix function of a Gaussian or elliptically contoured vector random field that is stationary, isotropic, and mean square continuous on the sphere is characterized.
Abstract: This article characterizes the covariance matrix function of a Gaussian or elliptically contoured vector random field that is stationary, isotropic, and mean square continuous on the sphere. By applying the characterization to examine the validity of a matrix function whose entries are polynomials of degrees up to 4, we obtain a necessary and sufficient condition for the polynomial matrix to be an isotropic covariance matrix function on the sphere.
TL;DR: In this paper, the authors prove integration by parts (IBP) formulas concerning maxima of solutions to some stochastic differential equations (SDEs) and deal with three types of maxima.
Abstract: In this article, we prove integration by parts (IBP) formulas concerning maxima of solutions to some stochastic differential equations (SDEs). We will deal with three types of maxima. First, we consider discrete time maximum, and then continuous time maximum in the case of one-dimensional SDEs. Finally, we deal with the maximum of the components of a solution to multi-dimensional SDEs. Applications to study their probability density functions by means of the IBP formulas are also discussed.
TL;DR: In this paper, a proof of well-posedness of an initial-boundary value problem involving a system of non-local parabolic partial differential equation (PDE) is presented.
Abstract: This article includes a proof of well posedness of an initial-boundary value problem involving a system of non-local parabolic partial differential equation (PDE), which naturally arises in the study of derivative pricing in a generalized market model, which is known as a semi-Markov modulated geometric Brownian motion (GBM) model We study the well posedness of the problem via a Volterra integral equation of second kind. A probabilistic approach, in particular the method of conditioning on stopping times is used for showing the uniqueness.
TL;DR: In this paper, the authors used the backward parametrix method to prove the existence and regularity of the transition density associated to the solution process of a stable-like driven stochastic differential equation with Holder continuous coefficients.
Abstract: In this article, we use the backward parametrix method in order to prove the existence and regularity of the the transition density associated to the solution process of a stable-like driven stochastic differential equation (SDE) with Holder continuous coefficients. The method of proof uses the parametrix method on the Gaussian component of a subordinated Brownian motion. This analysis which can be generalized also provides a stochastic representation of the density which is potentially useful for other applications.Abbrevations: B: Brownian motion; V: α-stable-like subordinator independent of B; μ: Levy measure of the subordinator V; m(·): positive concave increasing function; ; δy(dx): Dirac measure with unit mass at ; ψ: Levy exponent of Z; q(M, x): Gaussian density with covariance matrix M and ; ϕ: a regular varying function; b: drift coefficient of X; σ: coefficient of associated with the driving Levy process Z ≔ BV; ζ: coefficient associated with the diffusion (if X is a jump diffusion proce...
TL;DR: In this article, the transition densities of the basic affine jump-diffusion (BAJD) model with jumps were derived and shown to be continuous with respect to the Lebesgue measure.
Abstract: In this article, we find the transition densities of the basic affine jump-diffusion (BAJD), which has been introduced by Duffie and Gârleanu as an extension of the CIR model with jumps. We prove the positive Harris recurrence and exponential ergodicity of the BAJD. Furthermore, we prove that the unique invariant probability measure π of the BAJD is absolutely continuous with respect to the Lebesgue measure and we also derive a closed-form formula for the density function of π.
TL;DR: In this paper, anomalous diffusion on compact Riemannian manifolds, modeled by time-changed Brownian motions, is investigated, which is governed by equations involving the Laplace-Beltrami operator and a time-fractional derivative of order β ∈ (0, 1).
Abstract: We investigate anomalous diffusion on compact Riemannian manifolds, modeled by time-changed Brownian motions. These stochastic processes are governed by equations involving the Laplace–Beltrami operator and a time-fractional derivative of order β ∈ (0, 1). We also consider time dependent random fields that can be viewed as random fields on randomly varying manifolds.
TL;DR: In this article, an optimal hedging problem of the vulnerable European contingent claims is studied, where the underlying asset of the claims is assumed to be nontradable and the interest rate, the appreciation rate and the volatility of risky assets are modulated by a finite-state continuous-time Markov chain.
Abstract: This article focuses on an optimal hedging problem of the vulnerable European contingent claims. The underlying asset of the vulnerable European contingent claims is assumed to be nontradable. The interest rate, the appreciation rate and the volatility of risky assets are modulated by a finite-state continuous-time Markov chain. By using the local risk minimization method, we obtain an explicit closed-form solution for the optimal hedging strategies of the vulnerable European contingent claims. Further, we consider a problem of hedging for a vulnerable European call option. Optimal hedging strategies are obtained. Finally, a numerical example for the optimal hedging strategies of the vulnerable European call option in a two-regime case is provided to illustrate the sensitivities of the hedging strategies.
TL;DR: In this paper, the mean square stability and stabilization for stochastic delay systems with impulses were established using Razumikhin methodology, and two approaches, classical Lyapunov-based method and comparison principle, were proposed to develop sufficient conditions that guarantee the stability and stabilisation properties.
Abstract: This article establishes mean square stability and stabilization for stochastic delay systems with impulses. Using Razumikhin methodology, two approaches, classical Lyapunov-based method and comparison principle, are proposed to develop sufficient conditions that guarantee the stability and stabilization properties. It is shown that if the continuous system is stable and the impulses are destabilizing, the impulses should not be applied frequently. On the other hand, if the continuous system is unstable, but the impulses are stabilizing, the impulses should occur frequently to compensate the continuous state growth. Numerical examples are also presented to clarify the proposed theoretical results.
TL;DR: In this paper, the authors generalized the Beckner's type Poincare inequality to a large class of probability measures on an abstract Wiener space of the form μ⋆ν, where μ is the reference Gaussian measure and ν is a probability measure satisfying a certain integrability condition.
Abstract: We generalize the Beckner’s type Poincare inequality (Beckner, W. Proc. Amer. Math. Soc. (1989) 105:397–400) to a large class of probability measures on an abstract Wiener space of the form μ⋆ν, where μ is the reference Gaussian measure and ν is a probability measure satisfying a certain integrability condition. As the Beckner inequality interpolates between the Poincare and logarithmic Sobolev inequalities, we utilize a family of products for functions which interpolates between the usual point-wise multiplication and the Wick product. Our approach is based on the positivity of a quadratic form involving Wick powers and integration with respect to those convolution measures. In addition, we prove that in the finite-dimensional case the class of densities of convolutions measures satisfies a point-wise covariance inequality.
TL;DR: In this paper, the authors considered the blowup phenomenon of stochastic delayed evolution equations and established the sufficient condition to ensure the existence of a unique nonnegative solution for the problem of blow-up solutions in mean Lq-norm, q ⩾ 1, in a finite time.
Abstract: This article is concerned with the blowup phenomenon of stochastic delayed evolution equations. We first establish the sufficient condition to ensure the existence of a unique nonnegative solution of stochastic parabolic equations. Then the problem of blow-up solutions in mean Lq-norm, q ⩾ 1, in a finite time is considered. The main aim in this article is to investigate the effect of time delay and stochastic term. A new result shows that the stochastic delayed term can induce singularities.
TL;DR: This experiment measures the Hausdorff timing values, mean, standard deviation, and median of keystroke features, such as latency, duration, digraph, and their combinations, and compares their performance with the stochastic diffusion search for feature subset selection.
Abstract: Authentication plays an important role in dealing with security. Securing sensitive data and computer systems by allowing easy access for authenticated users and withstanding the attacks of imposters is one of the major challenges in the field of computer security. Nowadays, passwords have become the trend to control access to computer systems. Biometrics are used to measure and analyze an individual's unique behavioral or physiological patterns for authentication purposes. Keystroke dynamics have emerged as an important method in analyzing the typing rhythm in biometric techniques, as they provide an ease of use and increased trustworthiness associated with biometrics for creating username and password schemes. In this experiment, we measure the Hausdorff timing values, mean, standard deviation, and median of keystroke features, such as latency, duration, digraph, and their combinations, and compare their performance. The stochastic diffusion search is used for feature subset selection.
TL;DR: In this paper, the authors study kernel functions and associated reproducing kernel Hilbert spaces over infinite, discrete, and countable sets V and show how to find approximate solutions to boundary value problems using multiresolution-subdivision schemes in continuous domains.
Abstract: We study kernel functions, and associated reproducing kernel Hilbert spaces over infinite, discrete, and countable sets V. Numerical analysis builds discrete models (e.g., finite element) for the purpose of finding approximate solutions to boundary value problems; using multiresolution-subdivision schemes in continuous domains. In this article, we turn the tables: Our object of study is realistic infinite discrete models in their own right; and we then use an analysis of suitable continuous counterpart problems, but now serving as a tool for obtaining solutions in the discrete world.
TL;DR: The Gumbel test was first introduced by Lee and Mykland in 2008 from an economical point of view as discussed by the authors, where they considered a continuous-time stochastic volatility model with a general continuous volatility process and observed it under a high-frequency sampling scheme.
Abstract: This article gives an exhaustive mathematical analysis of the Gumbel test for additive jump components based on extreme value theory. The Gumbel test was first introduced by Lee and Mykland in 2008 from an economical point of view. They consider a continuous-time stochastic volatility model with a general continuous volatility process and observe it under a high-frequency sampling scheme. The test statistics based on the maximum of increments converges to the Gumbel distribution under the null hypothesis of no additive jump component and to infinity otherwise. Our article presents a moment method based technique that provides some deeper mathematical insights into the convergence and divergence case of the test statistics. In the non-jump case we are able to prove the convergence to the Gumbel distribution under greatly weak assumptions: The volatility process has to be merely pathwise Holder continuous with an arbitrary random Holder exponent and we have no restrictions concerning an additional d...
TL;DR: In this article, the authors consider a Markov chain generated by random iterations of a family of mappings indexed by elements of an arbitrary measurable space, and construct a set of place-dependent probability measures such that the chain converges to a stationary distribution.
Abstract: We consider a Markov chain generated by random iterations of a family of mappings indexed by elements of an arbitrary measurable space. Under sufficiently weak assumptions we construct a family of place-dependent probability measures such that considered Markov chain converges to a stationary distribution. We also prove some sufficient condition for asymptotic stability of a family of i.i.d. mappings and we apply obtained result for discrete white noise random dynamical systems showing analogous probabilistic long-time behavior.