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Showing papers in "Stochastic Analysis and Applications in 2013"


Journal ArticleDOI
TL;DR: In this article, the existence of a stochastic flow of Holder homeomorphisms for solutions of SDEs with singular time dependent drift having only certain integrability properties was proved.
Abstract: We prove the existence of a stochastic flow of Holder homeomorphisms for solutions of SDEs with singular time dependent drift having only certain integrability properties. We also show that the solution map x → X x is differentiable in a weak sense.

88 citations


Journal ArticleDOI
TL;DR: In this paper, the problem of global parameter estimation in the Cox-Ingersoll-Ross (CIR) model (X t ) t ≥ 0 was considered and new asymptotic results on the maximum likelihood estimator (MLE) associated to the global estimation of the drift parameters was established.
Abstract: This article deals with the problem of global parameter estimation in the Cox-Ingersoll-Ross (CIR) model (X t ) t≥0. This model is frequently used in finance for example, to model the evolution of short-term interest rates or as a dynamic of the volatility in the Heston model. In continuity with a recent work by Ben Alaya and Kebaier [1], we establish new asymptotic results on the maximum likelihood estimator (MLE) associated to the global estimation of the drift parameters of (X t ) t≥0. To do so, we need to study the asymptotic behavior of the quadruplet . This allows us to obtain various and original limit theorems on our MLE, with different rates and different types of limit distributions. Our results are obtained for both cases: ergodic and nonergodic diffusion.

58 citations


Journal ArticleDOI
TL;DR: In this paper, the existence and uniqueness of the strong pathwise solution of stochastic Navier-Stokes equation with Ito-Levy noise was studied and the recursive estimation of conditional expectation of the flow field given back measurements of sensor output data was formulated.
Abstract: In this article, we study the existence and uniqueness of the strong pathwise solution of stochastic Navier-Stokes equation with Ito-Levy noise. Nonlinear filtering problem is formulated for the recursive estimation of conditional expectation of the flow field given back measurements of sensor output data. The corresponding Fujisaki-Kallianpur-Kunita and Zakai equations describing the time evolution of the nonlinear filter are derived. Existence and uniqueness of measure-valued solutions are proven for these filtering equations.

38 citations


Journal ArticleDOI
TL;DR: In this article, some asymptotic formulas of the finite-time ruin probability for a two-dimensional renewal risk model are obtained, where the distributions of two claim amounts belong to the intersection of the long-tailed distributions class and the dominated varying distributions class, and the claim arrival times are extended negatively dependence structures.
Abstract: In this article, some asymptotic formulas of the finite-time ruin probability for a two-dimensional renewal risk model are obtained. In the model, the distributions of two claim amounts belong to the intersection of the long-tailed distributions class and the dominated varying distributions class and the claim arrival-times are extended negatively dependence structures. Assumption that the claim arrivals of two classes are governed by a common renewal counting process. The asymptotic formulas hold uniformly for t ∈ [f(x), ∞), where f(x) is an infinitely increasing function.

38 citations


Journal ArticleDOI
TL;DR: In this article, the Jacobian of the RDE flow driven by Gaussian signals is considered, and a user-friendly "transitivity property" of such integrability estimates is obtained.
Abstract: Integrability properties of (classical, linear, linear growth) rough differential equations (RDEs) are considered, the Jacobian of the RDE flow driven by Gaussian signals being a motivating example. We revisit and extend some recent ground-breaking work of Cass-Litterer-Lyons in this regard; as by-product, we obtain a user-friendly “transitivity property” of such integrability estimates. We also consider rough integrals; as a novel application, uniform Weibull tail estimates for a class of (random) rough integrals are obtained. A concrete example arises from the stochastic heat-equation, spatially mollified by hyper-viscosity, and we can recover (in fact, sharpen) a technical key result of Hairer [11].

36 citations


Journal ArticleDOI
TL;DR: This article proposes four types of covariance matrix structures for second-order Student's t vector random fields, which decay in power-law or log-law.
Abstract: This article deals with the Student's t vector random field, which is formulated as a scale mixture of Gaussian vector random fields, and whose finite-dimensional distributions decay in power-law and have heavy tails. There are two classes of Student's t vector random fields, one with second-order moments, and the other without a second-order moment. A Cauchy vector random field is an example of Student's t vector random fields without a first-order moment, and is also an example of Stable vector random fields. A second-order Student's t vector random field allows for any given correlation structure, just as a Gaussian vector random field does. We propose four types of covariance matrix structures for second-order Student's t vector random fields, which decay in power-law or log-law.

30 citations


Book ChapterDOI
TL;DR: In this article, a single server retrial queueing model, in which customers arrive according to a batch Markovian arrival process (BMAP), is considered, and steady state analysis of the model is performed.
Abstract: A single server retrial queueing model, in which customers arrive according to a batch Markovian arrival process (BMAP), is considered. An arriving batch, finding server busy, enters an orbit. Otherwise one customer from the arriving batch enters for service immediately while the rest join the orbit. The customers from the orbit try to reach the server subsequently and the inter-retrial times are exponentially distributed. Additionally, at each service completion epoch, two different search mechanisms are switched-on. Thus, when the server is idle, a competition takes place between primary customers, the customers coming by retrial and the two types of searches. It is assumed that if the type II search reaches the service facility ahead of the rest, all customers in the orbit are taken for service simultaneously, while in the other two cases, only a single customer is qualified to enter the service. We assume that the service times of the four types of customers namely, primary, repeated and those by the two types of searches are arbitrarily distributed with different distributions. Steady state analysis of the model is performed.

28 citations


Journal ArticleDOI
TL;DR: In this article, an optimal investment problem of an insurer in a hidden Markov, regime-switching, modeling environment using a backward stochastic differential equation (BSDE) approach is discussed.
Abstract: We discuss an optimal investment problem of an insurer in a hidden Markov, regime-switching, modeling environment using a backward stochastic differential equation (BSDE) approach. Filtering theory is used to transform the optimal investment problem into one with complete observations. Using BSDEs with jumps, we discuss the problem with complete observations.

22 citations


Journal ArticleDOI
TL;DR: In this paper, the authors investigated isotropic random fields for which the spectral density is unbounded at some frequencies and established limit theorems for weighted functionals of these random fields.
Abstract: The article investigates isotropic random fields for which the spectral density is unbounded at some frequencies. Limit theorems for weighted functionals of these random fields are established. It is shown that for a wide class of functionals, which includes the Donsker scheme, the limit is not affected by singularities at non-zero frequencies. For general schemes, in contrast to the Donsker line, we demonstrate that the singularities at non-zero frequencies play a role even for linear functionals.

21 citations


Journal ArticleDOI
TL;DR: In this article, a class of linear backward stochastic differential equations of descriptor type with time-invariant coefficients are introduced, and necessary and sufficient conditions for their solvability are obtained.
Abstract: In this article, a class of linear backward stochastic differential equations of descriptor type with time-invariant coefficients are introduced. Necessary and sufficient conditions for their solvability are obtained. It turns out that such equations may not always have a solution, and even when they do, some components of the solution could have a jump at terminal time. Exact controllability of linear descriptor stochastic control systems is also considered.

20 citations


Journal ArticleDOI
TL;DR: In this article, a general scaling setting in which Gaussian and non-Gaussian limit distributions of linear random fields can be obtained is defined, and the results derived cover the weak-dependence and strong-dependency cases for such Gaussian random fields.
Abstract: This article addresses the problem of defining a general scaling setting in which Gaussian and non-Gaussian limit distributions of linear random fields can be obtained. The linear random fields considered are defined by the convolution of a Green kernel, satisfying suitable scaling conditions, with a non-linear transformation of a Gaussian centered homogeneous random field. The results derived cover the weak-dependence and strong-dependence cases for such Gaussian random fields. Extension to more general random initial conditions defined, for example, in terms of non-linear transformations of χ2-random fields, is also discussed. For an example, we consider the random fractional diffusion equation. The vectorial version of the limit theorems derived is also formulated, including the limit distribution of the parabolically rescaled solution to the Burgers equation in the cases of weakly and strongly dependent initial potentials.

Journal ArticleDOI
TL;DR: In this article, the Polya-Aeppli process (PAP) is defined from three different points of view: as a compound Poisson process, as a delayed renewal process, and as a pure birth process.
Abstract: In this article, we study the Polya-Aeppli process (PAP). We define PAP from three different points of view: as a compound Poisson process, as a delayed renewal process, and as a pure birth process. We show that these definitions are equivalent. Also, using these definitions we identify several interesting characterizations of PAP.

Journal ArticleDOI
TL;DR: In this article, a new method of proving existence of weak solutions to stochastic differential equations with continuous coefficients having at most linear growth was developed, and the same method may be used even if the linear growth hypothesis is replaced with a suitable Lyapunov condition.
Abstract: In the first part of this article a new method of proving existence of weak solutions to stochastic differential equations with continuous coefficients having at most linear growth was developed. In this second part, we show that the same method may be used even if the linear growth hypothesis is replaced with a suitable Lyapunov condition.

Journal ArticleDOI
TL;DR: In this article, a splitting method for nonlinear stochastic equations of Schrodinger type is presented, which approximates the solution of the problem by the sequence of solutions of two types of equations.
Abstract: In this article, we construct a splitting method for nonlinear stochastic equations of Schrodinger type. We approximate the solution of our problem by the sequence of solutions of two types of equations: one without stochastic integral term, but containing the Laplace operator and the other one containing only the stochastic integral term. The two types of equations are connected to each other by their initial values. We prove that the solutions of these equations both converge strongly to the solution of the Schrodinger type equation.

Journal ArticleDOI
TL;DR: In this paper, the convergence of stochastic particle systems representing physical advection, inflow, outflow, and coagulation is studied on a bounded spatial domain such that there is a constant number of particles in the system.
Abstract: The convergence of stochastic particle systems representing physical advection, inflow, outflow, and coagulation is considered. The problem is studied on a bounded spatial domain such that there is...

Journal ArticleDOI
TL;DR: In this article, implicit one-step numerical schemes are presented for the pathwise simulation of stiff random ordinary differential equations (RODEs), specifically, implicit averaged Euler scheme and an implicit averaged midpoint scheme.
Abstract: Implicit one-step numerical schemes are presented for the pathwise simulation of stiff random ordinary differential equations (RODEs), specifically an implicit averaged Euler scheme and an implicit averaged midpoint scheme. These involve averaging the noise inside the vector field and lead to integrals of the noise over discretization subintervals. Distributions of such integrals for Wiener and Ornstein-Uhlenbeck processes are determined and enable their efficient simulation. Convergence and B-stability proofs are presented and the schemes are tested and compared for a three-compartment system arising in the modelling and treatment of the hepatitis C virus.

Journal ArticleDOI
Qian Lin1
TL;DR: In this article, the authors derived a general martingale characterization of G-Brownian motion, which generalizes the results obtained in Xu [17] for stochastic calculus with respect to G-martingales under sublinear expectation spaces.
Abstract: The objective of this article is to derive a general martingale characterization of G-Brownian motion, which generalizes the results obtained in Xu [17] For this end, we first study some extensions of stochastic calculus with respect to G-martingales under the sublinear expectation spaces

Journal ArticleDOI
TL;DR: In this article, a general mean convergence theorem for a normed sum of independent random variables is presented and some additional problems are posed, such as convergence in probability, mean convergence, and Kronecker lemma.
Abstract: Three examples are provided which demonstrate that “convergence in probability” versions of the Toeplitz lemma, the Cesaro mean convergence theorem, and the Kronecker lemma can fail. “Mean convergence” versions of the Toeplitz lemma, Cesaro mean convergence theorem, and the Kronecker lemma are presented and a general mean convergence theorem for a normed sum of independent random variables is established. Some additional problems are posed.

Journal ArticleDOI
TL;DR: In this paper, the nonlinear classical pure birth process and the fractional pure birth processes subordinated to various random times were analyzed and the state probability distribution was derived, and the corresponding governing differential equation was presented.
Abstract: We present and analyze the nonlinear classical pure birth process 𝒩(t), t > 0, and the fractional pure birth process 𝒩ν(t), t > 0, subordinated to various random times. We derive the state probability distribution , k ≥ 1 and, in some cases, we also present the corresponding governing differential equation. Various types of compositions of the fractional pure birth process 𝒩ν(t) have been examined in the second part of the paper. In particular, the processes 𝒩ν(T t ), 𝒩ν(𝒮α(t)), 𝒩ν(T 2ν(t)), have been analyzed, where T 2ν(t), t > 0, is a process related to fractional diffusion equations. As a byproduct of our analysis, some formulae relating Mittag–Leffler functions are obtained.

Journal ArticleDOI
TL;DR: In this article, the design and analysis of discrete-time Feynman-Kac particle integration models with geometric interacting jump processes with continuous time path integrals was studied, and nonasymptotic bias and variance theorems w.r.t.
Abstract: This article is concerned with the design and analysis of discrete time Feynman-Kac particle integration models with geometric interacting jump processes. We analyze two general types of model, corresponding to whether the reference process is in continuous or discrete time. For the former, we consider discrete generation particle models defined by arbitrarily fine time mesh approximations of the Feynman-Kac models with continuous time path integrals. For the latter, we assume that the discrete process is observed at integer times and we design new approximation models with geometric interacting jumps in terms of a sequence of intermediate time steps between the integers. In both situations, we provide nonasymptotic bias and variance theorems w.r.t. the time step and the size of the system, yielding what appear to be the first results of this type for this class of Feynman-Kac particle integration models. We also discuss uniform convergence estimates w.r.t. the time horizon. Our approach is based on an or...

Journal ArticleDOI
TL;DR: In this article, lower and upper bounds for the blowup times of a system of semilinear SPDEs were investigated under certain conditions on the system parameters, which allows us to use a formula due to Yor to obtain the distribution functions of several explosion times.
Abstract: We investigate lower and upper bounds for the blowup times of a system of semilinear SPDEs. Under certain conditions on the system parameters, we obtain explicit solutions of a related system of random PDEs, which allows us to use a formula due to Yor to obtain the distribution functions of several explosion times. We also give the Laplace transforms at independent exponential times of related exponential functionals of Brownian motion.

Journal ArticleDOI
TL;DR: In this paper, the first and second infinitesimal moments are estimated by using the local linear method of the underlying jump-diffusion models instead of standard kernel smoothing, and the estimators are consistent and asymptotically follow normal distribution under the condition of recurrence and stationarity.
Abstract: This article addresses the problem of nonparametric estimation of the first and second infinitesimal moments by using the local linear method of the underlying jump-diffusion models. The motivation behind the study is to use the asymmetric kernels instead of standard kernel smoothing. The basic idea relies on replacing the symmetric kernel by asymmetric kernel and provides a new way of obtaining the nonparametric estimation for jump-diffusion models. We prove that the estimators based on the local linear method for jump-diffusion models are consistent and asymptotically follow normal distribution under the condition of recurrence and stationarity.

Journal ArticleDOI
TL;DR: In this paper, the effective service time is modeled as a phase-type distribution, where a service can have at most a fixed number of interruptions and a super-clock will be set at the epoch of the first interruption and will be frozen whenever the service resumes from an interruption.
Abstract: In this article, we model an interrupted service scheme that has applications in practice using phase-type distribution. Services are subject to interruptions that occur according to a Poisson process. An interrupted service is either resumed from where left or repeated from the beginning based on a timer (referred to as a threshold clock) set at the epoch of an interruption. A service can have at most a fixed number of interruptions. Further, a super-clock will be set at the epoch of the first interruption and will be frozen whenever the service resumes from an interruption. However, if this super-clock expires before an interrupted service resumes, no further interruptions for this service can occur. Assuming all interruptions, threshold, and the super clocks as well as the service time to be of phase type (independent of each other), we show that the effective service time can be modeled as a phase-type distribution. Some illustrative numerical examples are presented.

Journal ArticleDOI
TL;DR: In this article, an inverse first-passage-time problem for Wiener process X(t) subject to random jumps from a boundary c is studied, where the problem consists of finding the distribution of the jumps which occur when X( t) hits c, so that the first passage time of X (t) through S has distribution F.
Abstract: We study an inverse first-passage-time problem for Wiener process X(t) subject to random jumps from a boundary c. Let be given a threshold S > X(0); and a distribution function F on [0, + ∞). The problem consists of finding the distribution of the jumps which occur when X(t) hits c, so that the first-passage time of X(t) through S has distribution F.

Journal ArticleDOI
TL;DR: In this paper, the authors study the duality of Markov processes in the sense that for a certain f, the generator of a Markov process can be seen as a dual process.
Abstract: The article is devoted to a study of the duality of processes in the sense that for a certain f. This classical topic has well known applications in interacting particles, intertwining, superprocesses, stochastic monotonicity, exit – entrance laws, ruin probabilities in finances, etc. Aiming mostly at the case of f depending on the difference of its arguments, we develop a systematic approach to duality via the analysis of the generators of dual Markov processes and illustrate this approach by various examples, in particular, by giving a full characterization of duality arising from Pareto order in R d .

Journal ArticleDOI
TL;DR: In this article, the concept of sigma-convergence associated to stochastic processes can be used to solve the homogenization problem of a nonlinear PDE.
Abstract: In this article, we show that the concept of sigma-convergence associated to stochastic processes can tackle the homogenization of stochastic partial differential equations In this regard, the homogenization of a stochastic nonlinear partial differential equation is addressed Using some deep compactness results such as the Prokhorov and Skorokhod theorems, we prove that the sequence of solutions of this problem converges in probability towards the solution of an equation of the same type To proceed with, we use the concept of sigma-convergence for stochastic processes, which takes into account both the deterministic and random behaviours of the solutions of the problem

Journal ArticleDOI
TL;DR: The existence of a notnecessarily unique strong solution for a stochastic differential equation with non-local sample dependence is established in this article under the assumption that the coefficients satisfy an asymptotically local boundedness condition in addition to continuity.
Abstract: The existence of a not-necessarily-unique strong solution for a stochastic differential equations with nonlocal sample dependence is established under the assumption that the coefficients satisfy an asymptotically local boundedness condition in addition to continuity. The proof is by an Euler-like construction of approximations. These equations include mean-field stochastic differential equations, but the nonlocal sample dependence can be more general than just the dependence on moments of the solution.

Journal ArticleDOI
TL;DR: Bertoin and Yor as discussed by the authors showed that positive self-similar Markov processes (pssMps) that only jump downwards and do not hit zero in finite time are uniquely determined by their entire moments for which explicit formulas have been derived.
Abstract: Bertoin and Yor [2] have shown that the law of positive self-similar Markov processes (pssMps) that only jump downwards and do not hit zero in finite time are uniquely determined by their entire moments for which explicit formulas have been derived. We use a recent jump-type stochastic differential equation approach to reprove and to extend their formulas.

Journal ArticleDOI
TL;DR: In this paper, a partial reverse of this theorem is proved: if X 1, X, X n are independent infinitely divisible random variables such that has the chi-square distribution with n degrees of freedom then random variables X 1.
Abstract: Recall that if independent random variables X 1,…, X n have the same standard normal distribution then has the chi-square distribution with n degrees of freedom. In this note, a partial reverse of this theorem is proved: if X 1,…, X n are independent infinitely divisible random variables such that has the chi-square distribution with n degrees of freedom then random variables X 1,…, X n have the same standard normal distribution.

Journal ArticleDOI
TL;DR: In this paper, a robust guaranteed cost control of stochastic discrete-time systems with parametric uncertainties under Markovian switching is considered, where the control is simultaneously applied to both the random and the deterministic components of the system.
Abstract: A problem of robust guaranteed cost control of stochastic discrete-time systems with parametric uncertainties under Markovian switching is considered. The control is simultaneously applied to both the random and the deterministic components of the system. The noise (the random) term depends on both the states and the control input. The jump Markovian switching is modeled by a discrete-time Markov chain and the noise or stochastic environmental disturbance is modeled by a sequence of identically independently normally distributed random variables. Using linear matrix inequalities (LMIs) approach, the robust quadratic stochastic stability is obtained. The proposed control law for this quadratic stochastic stabilization result depended on the mode of the system. This control law is developed such that the closed-loop system with a cost function has an upper bound under all admissible parameter uncertainties. The upper bound for the cost function is obtained as a minimization problem. Two numerical examples a...