Showing papers in "Stochastic Analysis and Applications in 2014"
TL;DR: In this article, the authors construct compositions of vector processes of the form, t > 0,, β ∈ (0, 1),, whose distribution is related to space-time fractional n-dimensional telegraph equations.
Abstract: In this work, we construct compositions of vector processes of the form , t > 0, , β ∈ (0, 1], , whose distribution is related to space-time fractional n-dimensional telegraph equations. We present within a unifying framework the pde connections of n-dimensional isotropic stable processes S2βn whose random time is represented by the inverse , t > 0, of the superposition of independent positively skewed stable processes, , t > 0, (H2ν1, Hν2, independent stable subordinators). As special cases for n = 1, and β = 1, we examine the telegraph process T at Brownian time B ([14]) and establish the equality in distribution , t > 0. Furthermore the iterated Brownian motion ([2]) and the two-dimensional motion at finite velocity with a random time are investigated. For all these processes, we present their counterparts as Brownian motion at delayed stable-distributed time.
55 citations
Journal Article•
TL;DR: In this article, the authors proposed a numerical method to approximate the solution of a Backward Stochastic Differential Equations with Jumps (BSDEJ) based on the construction of a discrete BSDEJ driven by a complete system of three orthogonal discrete time-space martingales.
Abstract: In this paper we propose a numerical method to approximate the solution of a Backward Stochastic Differential Equations with Jumps (BSDEJ). This method is based on the construction of a discrete BSDEJ driven by a complete system of three orthogonal discrete time-space martingales, the first a random walk converging to a Brownian motion; the second, another random walk, independent of the first one, converging to a Poisson process. The solution of this discrete BSDEJ is shown to weakly converge to the solution of the continuous time BSDEJ. An application to partial integro-differential equations is given.
30 citations
TL;DR: This work provides theoretical results which quantify, in terms of ε, the ABC error in approximation of expectations of additive functionals with respect to the smoothing distributions and forms some of the first quantitative results for ABC methods which jointly treat the ABC and simulation errors.
Abstract: We consider a method for approximate inference in hidden Markov models (HMMs). The method circumvents the need to evaluate conditional densities of observations given the hidden states. It may be considered an instance of Approximate Bayesian Computation (ABC) and it involves the introduction of auxiliary variables valued in the same space as the observations. The quality of the approximation may be controlled to arbitrary precision through a parameter e > 0. We provide theoretical results which quantify, in terms of e, the ABC error in approximation of expectations of additive functionals with respect to the smoothing distributions. Under regularity assumptions, this error is , where n is the number of time steps over which smoothing is performed. For numerical implementation, we adopt the forward-only sequential Monte Carlo (SMC) scheme of [14] and quantify the combined error from the ABC and SMC approximations. This forms some of the first quantitative results for ABC methods which jointly treat the AB...
28 citations
TL;DR: This article considers a linear-quadratic optimal control problem (LQ problem) for a controlled linear stochastic differential equation driven by a multidimensional Browinan motion and a Poisson random martingale measure in the general case, where the coefficients are allowed to be predictable processes or random matrices.
Abstract: In this article, we consider a linear-quadratic optimal control problem (LQ problem) for a controlled linear stochastic differential equation driven by a multidimensional Browinan motion and a Poisson random martingale measure in the general case, where the coefficients are allowed to be predictable processes or random matrices. By the duality technique, the dual characterization of the optimal control is derived by the optimality system (so-called stochastic Hamilton system), which turns out to be a linear fully coupled forward-backward stochastic differential equation with jumps. Using a decoupling technique, the connection between the stochastic Hamilton system and the associated Riccati equation is established. As a result, the state feedback representation is obtained for the optimal control. As the coefficients for the LQ problem are random, here, the associated Riccati equation is a highly nonlinear backward stochastic differential equation (BSDE) with jumps, where the generator depends on the unkn...
27 citations
TL;DR: In this article, the authors considered a deterministic dynamical system with the global attractor 𝒜 which supports a unique ergodic probability measure P. The measure P can be considered as the uniform long-term mean of the trajectories staying in a bounded domain D containing &#d 49c; and solved the first exit time and location problem from D in the limit of ϵ↘0.
Abstract: We consider a finite-dimensional deterministic dynamical system with the global attractor 𝒜 which supports a unique ergodic probability measure P. The measure P can be considered as the uniform long-term mean of the trajectories staying in a bounded domain D containing 𝒜. We perturb the dynamical system by a multiplicative heavy tailed Levy noise of small intensity ϵ > 0 and solve the asymptotic first exit time and location problem from D in the limit of ϵ↘0. In contrast to the case of Gaussian perturbations, the exit time has an algebraic exit rate as a function of ϵ, just as in the case when 𝒜 is a stable fixed point studied earlier in [9, 14, 19, 26]. As an example, we study the first exit problem from a neighborhood of the stable limit cycle for the Van der Pol oscillator perturbed by multiplicative α-stable Levy noise.
25 citations
TL;DR: In this article, a large deviation principle is built for the following singularly perturbed stochastic nonlinear damped wave equations on bounded regular domains, where the small parameter ν parametrises both the strength of noise and the singular perturbation.
Abstract: A large deviation principle is built for the following singularly perturbed stochastic nonlinear damped wave equations on bounded regular domains: We use a weak convergence method. The small parameter ν parametrises both the strength of noise and the singular perturbation. The rate function of large deviations is proven to be that of the large deviations for the stochastic heat equation This result shows the effectiveness of asymptotic approximation of the stochastic heat equation to singularly perturbed stochastic wave equations.
22 citations
TL;DR: In this article, the authors derived derivative formulae and Harnack inequalities for linear stochastic differential equations driven by Levy processes, using lower bound conditions of the Levy measure.
Abstract: By using lower bound conditions of the Levy measure, derivative formulae and Harnack inequalities are derived for linear stochastic differential equations driven by Levy processes. As applications, explicit gradient estimates and heat kernel inequalities are presented. As byproduct, a new Girsanov theorem for Levy processes is derived.
20 citations
TL;DR: In this article, the authors prove the existence and uniqueness of quasi-stationary distributions for symmetric Markov processes with intrinsic ultracontractive Markov semigroup, and apply their results to one-dimensional diffusion processes.
Abstract: We prove the existence and uniqueness of quasi-stationary distributions for symmetric Markov processes. In particular, we show that if its Markov semigroup is intrinsic ultracontractive, then there exists a unique quasi-stationary distribution. We apply our results to one-dimensional diffusion processes.
19 citations
TL;DR: In this article, the qualitative dynamics of an SIRS epidemic model with saturated incidence in random environments were examined, and it was shown that the global positive solutions of the stochastic model belong to a bounded positively invariant set with probability one.
Abstract: This article examines the qualitative dynamics of an SIRS epidemic model with saturated incidence in random environments. First, we show that the global positive solutions of the stochastic model belongs to a bounded positively invariant set with probability one. Then we obtain the threshold between persistence and extinction of diseases. Furthermore, we establish certain asymptotic results regarding large time behavior. The results show that the introduction of noise in the deterministic SIRS model can change the properties of the epidemic model significantly.
18 citations
TL;DR: In this article, the averaging principle for multivalued stochastic differential equations (MSDEs) driven by Brownian motion with Brownian noise is investigated, and their solutions are quantitatively compared.
Abstract: The averaging principle for multivalued stochastic differential equations (MSDEs) driven by Brownian motion with Brownian noise is investigated. An averaged MSDEs for the original MSDEs is proposed, and their solutions are quantitatively compared. Under suitable assumptions, it is shown that the solution of the MSDEs converges to that of the original MSDEs in the sense of mean square and also in probability. Two examples are presented to illustrate the averaging principle.
17 citations
TL;DR: In this article, a Feller-type diffusion approximation is derived for critical multi-type branching processes with immigration when the offspring mean matrix is primitive (in other words, positively regular).
Abstract: Under natural assumptions a Feller-type diffusion approximation is derived for critical multi-type branching processes with immigration when the offspring mean matrix is primitive (in other words, positively regular). Namely, it is proved that a sequence of appropriately scaled random step functions formed from a sequence of critical primitive multi-type branching processes with immigration converges weakly toward a squared Bessel process supported by a ray determined by the Perron vector of the offspring mean matrix.
TL;DR: In this paper, the problem of the definition and computation of an H2-type norm for discrete-time time-varying periodic stochastic linear systems simultaneously affected by multiplicative white noise perturbations and random jumping according to a Markov chain with an infinite countable number of states was investigated.
Abstract: This article investigates the problem of the definition and computation of an H2-type norm for discrete-time time-varying periodic stochastic linear systems simultaneously affected by multiplicative white noise perturbations and random jumping according to a Markov chain with an infinite countable number of states. Also, we solve an optimization problem that contains, as a special case, the H2 optimal control problem for the considered class of stochastic systems under the assumption of perfect state measurements.
TL;DR: For stochastic differential equation driven by fractional Brownian motion with Hurst parameter H > 1/2, Harnack-type inequalities are established by constructing a coupling with unbounded time-dependent drift.
Abstract: For stochastic differential equation driven by fractional Brownian motion with Hurst parameter H > 1/2, Harnack-type inequalities are established by constructing a coupling with unbounded time-dependent drift. These inequalities are applied to the study of existence and uniqueness of invariant measure for a discrete Markov semigroup constructed in terms of the distribution of the solution. Furthermore, we show that entropy-cost inequality holds for the invariant measure.
TL;DR: In this paper, the authors established sufficient conditions ensuring almost sure practical asymptotic stability with a nonexponential decay rate for solutions to stochastic evolution equations based on Lyapunov techniques.
Abstract: In this article we establish some sufficient conditions ensuring almost sure practical asymptotic stability with a non-exponential decay rate for solutions to stochastic evolution equations based on Lyapunov techniques.
TL;DR: In this article, the authors derived pseudoprocesses whose law is governed by heat-type equations of real-valued order γ > 2, where the heat type equations involve either Riesz operators or their Feller asymmetric counterparts.
Abstract: In this article, we construct pseudo random walks (symmetric and asymmetric) that converge in law to compositions of pseudoprocesses stopped at stable subordinators. We find the higher-order space-fractional heat-type equations whose fundamental solutions coincide with the law of the limiting pseudoprocesses. The fractional equations involve either Riesz operators or their Feller asymmetric counterparts. The main result of this article is the derivation of pseudoprocesses whose law is governed by heat-type equations of real-valued order γ > 2. The classical pseudoprocesses are very special cases of those investigated here.
TL;DR: In this article, the inverse first-hitting problem for a one-dimensional, time-homogeneous diffusion X(t) reflected between two boundaries a and b, which starts from a random position η, was studied.
Abstract: We study an inverse first-hitting problem for a one-dimensional, time-homogeneous diffusion X(t) reflected between two boundaries a and b, which starts from a random position η. Let a ≤ S ≤ b be a given threshold, such that P(η e [a, S]) = 1, and F an assigned distribution function. The problem consists of finding the distribution of η such that the first-hitting time of X to S has distribution F. This is a generalization of the analogous problem for ordinary diffusions, that is, without reflecting, previously considered by the author.
TL;DR: In this paper, the strong convergence of Euler approximations of stochastic delay differential equations is proved under general conditions, and the rate of convergence is obtained under one-sided and polynomial Lipschitz conditions.
Abstract: The strong convergence of Euler approximations of stochastic delay differential equations is proved under general conditions. The assumptions on drift and diffusion coefficients have been relaxed to include polynomial growth and only continuity in the arguments corresponding to delays. Furthermore, the rate of convergence is obtained under one-sided and polynomial Lipschitz conditions. Finally, our findings are demonstrated with the help of numerical simulations.
TL;DR: In this article, the translation theorem was used to obtain relations involving integral transforms and convolution products in quantum mechanics, and several useful formulas involving various functionals, which arise naturally in quantum physics.
Abstract: In this article, we use the translation theorem to obtain several relationships involving integral transforms and convolution products. In particular, we obtain several useful formulas involving various functionals, which arise naturally in quantum mechanics.
TL;DR: In this article, the authors give an overview of maximal inequalities and limit theorems for the tail probabilities for the supremum of a fractional Brownian motion, and give a general overview of some maximal inequalities.
Abstract: We give an overview of some maximal inequalities and limit theorems for the tail probabilities for the supremum of a fractional Brownian motion.
TL;DR: In this paper, a survey of recent results on invariances and ergodicity of general quantum Markov semigroups of bounded linear maps acting on C*-or von Neumann algebra is presented.
Abstract: This article surveys recent results on invariances and ergodicity of general quantum Markov semigroups of bounded linear maps acting on C*- or von Neumann algebra . In particular, we consider existence and uniqueness of invariant (stationary) quantum states as well as ergodicity and mean ergodicity of quantum states via heavy usage of the GNS representation. This survey is made self-contained by also reviewing relevant concepts and results necessary for the subsequent developments.
TL;DR: In this article, a Donsker-type approximation of the fractional Brownian motion was established for weak convergence of fractional Wiener integrals, where the convergence was proved for Volterra Gaussian processes.
Abstract: We prove a Donsker-type approximation of the fractional Brownian motion which extends a result by Sottinen for the case H > 1/2 to the full range of Hurst parameters H ∈ (0, 1). The convergence is established by a Donsker-type theorem for Volterra Gaussian processes. The approximation is applied to weak convergence of fractional Wiener integrals.
TL;DR: In this article, a sufficient stochastic maximum principle for the optimal control of a semi-Markov modulated jump-diffusion process in which the drift, diffusion, and the jump kernel of the jump diffusion process is modulated by a semiautomatic process is provided.
Abstract: The finite state semi-Markov process is a generalization over the Markov chain in which the sojourn time distribution is any general distribution. In this article, we provide a sufficient stochastic maximum principle for the optimal control of a semi-Markov modulated jump-diffusion process in which the drift, diffusion, and the jump kernel of the jump-diffusion process is modulated by a semi-Markov process. We also connect the sufficient stochastic maximum principle with the dynamic programming equation. We apply our results to finite horizon risk-sensitive control portfolio optimization problem and to a quadratic loss minimization problem.
TL;DR: In this paper, the authors derived the piecewise Lyapunov-Krasovskii functional, delay-dependent stability conditions by using freeweighting matrix and convex combination approach.
Abstract: In this article, the problem of stochastic stability analysis for switched stochastic genetic regulatory networks with interval time-varying delays based on average dwell time approach is investigated. By constructing the piecewise Lyapunov-Krasovskii functional, delay-dependent stability conditions are derived by using free-weighting matrix and convex combination approach. The derived stability conditions are expressed in terms of linear matrix inequalities which can be easily solved by using the MATLAB LMI control toolbox. Finally, numerical examples are provided to demonstrate the effectiveness and less conservativeness of the proposed theoretical results.
TL;DR: In this paper, it was shown that the transition semigroup (P t ) t = 0 corresponding to stochastic dissipative equations driven by α-stable is strong Feller and irreducible for α ∈ (1, 2).
Abstract: It is shown that the transition semigroup (P t ) t≥0 corresponding to stochastic dissipative equations driven by α-stable is strong Feller and irreducible for α ∈ (1, 2). This result ensures the ergodicity for the equation.
TL;DR: This paper showed that the known examples of diverse markets in the literature can be approximated uniformly (on the logarithmic scale) by models that are both diverse and arbitrage-free.
Abstract: A stock market is called diverse if no stock can dominate the market in terms of relative capitalization. On one hand, this natural property leads to arbitrage in diffusion models under mild assumptions. On the other hand, it is also easy to construct diffusion models which are both diverse and free of arbitrage. Can one tell whether an observed diverse market admits arbitrage? In this article, we argue that this may well be impossible by proving that the known examples of diverse markets in the literature (which do admit arbitrage) can be approximated uniformly (on the logarithmic scale) by models that are both diverse and arbitrage-free.
TL;DR: The Kolmogorov-Feller WLLN is extended to AANA sequences which are strictly weaker than negatively associated sequences by modifying the arguments of Kruglov [12].
Abstract: The Kolmogorov-Feller WLLN is extended to AANA sequences which are strictly weaker than negatively associated sequences by modifying the arguments of Kruglov [12]. The assumption of identical distribution has been dispensed with in a natural way.
TL;DR: In this paper, the authors study controllability properties for linear stochastic systems of mean-field type and provide necessary and sufficient criteria for exact terminal-controllability.
Abstract: We study some controllability properties for linear stochastic systems of mean-field type. First, we give necessary and sufficient criteria for exact terminal-controllability. Second, we characterize the approximate and approximate null-controllability via duality techniques. Using Riccati equations associated to linear quadratic problems in the control of mean-field systems, we provide a (conditional) viability criterion for approximate null-controllability. In the classical diffusion framework, approximate and approximate null-controllability are equivalent. This is no longer the case for mean-field systems. We provide sufficient (algebraic) invariance conditions implying approximate null-controllability. We also present a general class of systems for which our criterion is equivalent to approximate null-controllability property. We also introduce some rank conditions under which approximate and approximate null-controllability are equivalent. Several examples and counter-examples as well as a partial a...
TL;DR: In this paper, a splitting-up scheme for the numerical approximation of the 3D stochastic Navier-Stokes-α model is proposed and proved to converge to the unique variational solution when the time step tends to zero.
Abstract: We propose and analyze a splitting-up scheme for the numerical approximation of the 3D stochastic Navier-Stokes-α model. We prove the convergence of the scheme to the unique variational solution of the 3D stochastic Navier-Stokes-α model when the time step tends to zero.
TL;DR: In this article, the authors considered iterated function systems with place-dependent probabilities and provided assumptions ensuring the equicontinuity of iteration orbits, where P n f: n ∈ ℕ is the dual of transition operator corresponding to an IFS.
Abstract: Iterated function systems with place-dependent probabilities are considered in this article. Assumptions ensuring the equicontinuity of iteration orbits {P n f: n ∈ ℕ} of bounded continuous functions f, where P is the dual of transition operator corresponding to an IFS, are presented. The results are applied in the study of asymptotic stability properties of these systems.
TL;DR: The biology of phages is reviewed, models of phage are discussed, and new stochastic models are formed based on an underlying ordinary differential equation model to study the impact of variability on phage therapy.
Abstract: Phages are viruses that kill bacteria and are used to treat bacterial infections. We review the biology of phage, discuss models of phage, then formulate new stochastic models with the goal of studying the impact of variability on phage therapy. Two new stochastic models are derived based on an underlying ordinary differential equation model. A continuous-time Markov chain model and a stochastic differential equation model account for variability due to adsorption, reproduction and release of phage particles. A branching process approximation yields an estimate for the probability of phage extinction. Numerical examples highlight the importance of variability in modeling phage.