Showing papers in "Stochastic Analysis and Applications in 2015"
TL;DR: A particle filter for approximating Feynman–Kac models with indicator potentials is investigated and a newly developed special case of the locally adaptive particle filter is used that can deal with this latter problem, while introducing a random cost per-time step.
Abstract: In the following article, we investigate a particle filter for approximating Feynman–Kac models with indicator potentials and we use this algorithm within Markov chain Monte Carlo (MCMC) to learn static parameters of the model. Examples of such models include approximate Bayesian computation (ABC) posteriors associated with hidden Markov models (HMMs) or rare-event problems. Such models require the use of advanced particle filter or MCMC algorithms to perform estimation. One of the drawbacks of existing particle filters is that they may “collapse,” in that the algorithm may terminate early, due to the indicator potentials. In this article, using a newly developed special case of the locally adaptive particle filter, we use an algorithm that can deal with this latter problem, while introducing a random cost per-time step. In particular, we show how this algorithm can be used within MCMC, using particle MCMC. It is established that, when not taking into account computational time, when the new MCMC algorith...
39 citations
TL;DR: In this article, the existence of mild solutions to stochastic impulsive evolution equations with time delays, driven by fractional Brownian motion with the Hurst index H > 1/2 via a new fixed point analysis approach was studied.
Abstract: In this article, we study the existence of mild solutions to stochastic impulsive evolution equations with time delays, driven by fractional Brownian motion with the Hurst index H > 1/2 via a new fixed point analysis approach.
27 citations
TL;DR: In this article, a class of stochastic SIRS-type epidemic models with nonlinear incidence was investigated, and the existence of the unique global positive solution with any positive initial value was established.
Abstract: In this article, we investigate a class of stochastic SIRS-type epidemic models with nonlinear incidence. The threshold value is identified. The existence of the unique global positive solution with any positive initial value is established. The sufficient conditions for the extinction of the disease and the permanence in the mean of the model with probability one are also established. That is, if , then the disease is extinct with probability one under certain parametric restrictions, and if , then the model is permanent in the mean with probability one. Furthermore, when there is not the disease-related death, if , then the existence of unique stationary distribution of the model is obtained.
27 citations
TL;DR: In this article, a law of large numbers for the power variation of an integrated fractional process in a pure jump model has been proved, which yields consistency of an estimator for the integrated volatility where we are no longer restricted to a Gaussian model.
Abstract: We prove a law of large numbers for the power variation of an integrated fractional process in a pure jump model. This yields consistency of an estimator for the integrated volatility where we are no longer restricted to a Gaussian model.
25 citations
TL;DR: In this article, the authors prove the existence and a characterization of optimal risk-sensitive control under geometric ergodicity of the state dynamics along with a smallness condition on the running cost.
Abstract: In this article, we study risk-sensitive control problem with controlled continuous time Markov chain state dynamics. Using multiplicative dynamic programming principle along with the atomic structure of the state dynamics, we prove the existence and a characterization of optimal risk-sensitive control under geometric ergodicity of the state dynamics along with a smallness condition on the running cost.
25 citations
TL;DR: In this article, the Doleans-Dade exponential of a stochastic integral with respect to a counting process local martingale is defined and sufficient criteria for the construction of non-explosive Hawkes processes, counting processes depending on diffusion processes as well as inhomogeneous finite-state Markov processes.
Abstract: We give sufficient criteria for the Doleans-Dade exponential of a stochastic integral with respect to a counting process local martingale to be a true martingale. The criteria are adapted particularly to the case of counting processes and are sufficiently weak to be useful and verifiable, as we illustrate by several examples. In particular, the criteria allow for the construction of for example nonexplosive Hawkes processes, counting processes with stochastic intensities depending on diffusion processes as well as inhomogeneous finite-state Markov processes.
23 citations
TL;DR: In this article, the authors studied the existence and uniqueness of mild solutions, Yosida approximations, and the weak convergence of induced probability measures for a semilinear McKean-Vlasov type stochastic evolution equation in a real Hilbert space.
Abstract: This article is concerned with a semilinear McKean–Vlasov type stochastic evolution equation in a real Hilbert space. The main goal of the article is to study the existence and uniqueness of mild solutions, Yosida approximations of mild solutions of such equations, and to deduce the weak convergence of the corresponding induced probability measures. As an application, we also study the exponential stability of mild solutions of such equations.
22 citations
TL;DR: In this paper, a fractional generalization of the well-known Gamma process is studied, in which the corresponding densities satisfy the same differential equation as the usual Gamma process, but with the shift operator replaced by its fractional version of order ν > 0.
Abstract: We introduce and study fractional generalizations of the well-known Gamma process, in the following sense: the corresponding densities are proved to satisfy the same differential equation as the usual Gamma process, but with the shift operator replaced by its fractional version of order ν > 0 In the case ν > 1, the solution corresponds to the density of a Gamma process time-changed by an independent stable subordinator of index 1/ν For ν less than one an analogous result holds, with the subordinator replaced by the inverse In this case the fractional Gamma process is proved to be a non-stationary version of the standard one, with power law behavior of the expected value Hence it can be considered a useful tool in modelling stochastic deterioration in the non-linear cases, a situation which often occurs in real data (see ie, [42] and the references therein)As a consequence of the previous results, the fractional generalizations of some Gamma subordinated processes (ie the Variance Gamma, the Geome
21 citations
TL;DR: In this paper, the authors considered related bilinear forms that generalize the energy form for a particle in an electromagnetic field, and obtained one by semigroup approximation and another, closed one, by using a Feynman-Kac-Ito formula.
Abstract: Given a (conservative) symmetric Markov process on a metric space we consider related bilinear forms that generalize the energy form for a particle in an electromagnetic field. We obtain one bilinear form by semigroup approximation and another, closed one, by using a Feynman–Kac–Ito formula. If the given process is Feller, its energy measures have densities and its jump measure has a kernel, then the two forms agree on a core and the second is a closed extension of the first. In this case we provide the explicit form of the associated Hamiltonian.
20 citations
TL;DR: In this article, the authors provide strong law of large numbers and central limit theorem results for inhomogeneous Markov chains with respect to a continuous-time Markov chain model.
Abstract: Limit theorems for functionals of classical (homogeneous) Markov renewal and semi-Markov processes have been known for a long time, since the pioneering work of Pyke Schaufele (Limit theorems for Markov renewal processes, Ann. Math. Statist., 35(4):1746–1764, 1964). Since then, these processes, as well as their time-inhomogeneous generalizations, have found many applications, for example, in finance and insurance. Unfortunately, no limit theorems have been obtained for functionals of inhomogeneous Markov renewal and semi-Markov processes as of today, to the best of the authors’ knowledge. In this article, we provide strong law of large numbers and central limit theorem results for such processes. In particular, we make an important connection of our results with the theory of ergodicity of inhomogeneous Markov chains. Finally, we provide an application to risk processes used in insurance by considering a inhomogeneous semi-Markov version of the well-known continuous-time Markov chain model, widely used in...
15 citations
TL;DR: By using coupling by change of measures, the Driver-type integration by parts formula was established for a class of stochastic differential equations driven by fractional Brownian motions in this paper.
Abstract: By using coupling by change of measures, the Driver-type integration by parts formula is established for a class of stochastic differential equations driven by fractional Brownian motions. As applications, (log) shift Harnack inequalities and estimates on the distribution density of the solutions are presented.
TL;DR: In this paper, the authors studied processes that are constructed by a convolution of a deterministic kernel with a martingale and derived maximal inequalities in terms of the kernel and of the moments of the driving Martingale.
Abstract: In this article we study processes that are constructed by a convolution of a deterministic kernel with a martingale. A special emphasis is put on the case where the driving martingale is a centred Levy process, which covers the popular class of fractional Levy processes. As a main result we show that, under appropriate assumptions on the kernel and the martingale, the maximum process of the corresponding “convoluted martingale” is p-integrable and we derive maximal inequalities in terms of the kernel and of the moments of the driving martingale.
TL;DR: In this paper, the authors studied the long time behavior of a discrete time weakly interacting particle system, and the corresponding nonlinear Markov process in, described in terms of a general stochastic evolution equation.
Abstract: We study long time behavior of a discrete time weakly interacting particle system, and the corresponding nonlinear Markov process in , described in terms of a general stochastic evolution equation. In a setting where the state space of the particles is compact such questions have been studied in previous works, however, for the case of an unbounded state space very few results are available. Under suitable assumptions on the problem data we study several time asymptotic properties of the N-particle system and the associated nonlinear Markov chain. In particular, we show that the evolution equation for the law of the nonlinear Markov chain has a unique fixed point and starting from an arbitrary initial condition convergence to the fixed point occurs at an exponential rate. The empirical measure μNn of the N-particles at time n is shown to converge to the law μn of the nonlinear Markov process at time n, in the Wasserstein-1 distance, in L1, as N → ∞, uniformly in n. Several consequences of this uniform con...
TL;DR: In this paper, the authors investigated a Schrodinger problem with multiplicative Gaussian noise term and power-type nonlinearity on a bounded one-dimensional domain and proved the existence and uniqueness of the variational solution.
Abstract: We investigate a Schrodinger problem with multiplicative Gaussian noise term and power-type nonlinearity on a bounded one-dimensional domain. In order to prove the existence and uniqueness of the variational solution, a further process will be introduced which allows to transform the stochastic nonlinear Schrodinger problem into a pathwise one. Galerkin approximations and compact embedding results are used.
TL;DR: In this article, the authors introduce concepts and surveys recent results on recurrence and transience of general quantum Markov semigroups (QMS) of bounded linear maps acting on a C*-or von Neumann algebra.
Abstract: This article introduces concepts and surveys recent results on recurrence and transience of general quantum Markov semigroups (QMS) of bounded linear maps acting on a C*- or von Neumann algebra . In particular, the concept of potentials for classical Markov semigroups/processes is extended to its noncommutative counterpart. The characterization of recurrent and transient quantum Markov semigroups and classification of irreducible quantum Markov semigroups are established in terms of the potential of some subharmonic projection for the QMS. This introductory and survey work can be treated as a continuation of the closely related paper by Chang [12], which dealt with the invariance, mean ergodicity and ergodicity of QMS. Since it is intended as an introduction to large time asymptotic behavior of quantum Markov semigroups, this article is made self-contained by reviewing relevant concepts and results in quantum probability space, quantum states, and quantum Markov semigroups that are necessary for the subse...
TL;DR: In this article, the authors studied the rate of convergence of the weak Euler approximation for Ito diffusion and jump processes with Holder-continuous generators and showed that the Euler scheme yields a positive weak order of convergence.
Abstract: This article studies the rate of convergence of the weak Euler approximation for Ito diffusion and jump processes with Holder-continuous generators. It covers a number of stochastic processes including the nondegenerate diffusion processes and a class of stochastic differential equations driven by stable processes. To estimate the rate of convergence, the existence of a unique solution to the corresponding backward Kolmogorov equation in Holder space is first proved. It then shows that the Euler scheme yields positive weak order of convergence.
TL;DR: In this paper, the convergence in probability in the non-standard M1 Skorokhod topology of the Hilbert valued stochastic convolution integrals of the type to a process driven by a Levy process L was studied.
Abstract: We study the convergence in probability in the non-standard M1 Skorokhod topology of the Hilbert valued stochastic convolution integrals of the type to a process driven by a Levy process L. In Banach spaces, we introduce strong, weak. and product modes of -convergence, prove a criterion for the -convergence in probability of stochastically continuous cadlag processes in terms of the convergence in probability of the finite dimensional marginals and a good behavior of the corresponding oscillation functions, and establish criteria for the convergence in probability of Levy driven stochastic convolutions. The theory is applied to the infinitely dimensional integrated Ornstein–Uhlenbeck processes with diagonalizable generators.
TL;DR: In this article, the authors considered set-valued stochastic integrals driven by two-parameter martingales and increasing processes and established the existence and uniqueness of solutions to such equations as well as their additional properties.
Abstract: This article is concerned with notions of set-valued stochastic integrals driven by two-parameter martingales and increasing processes. We investigate their main properties and we consider next multivalued stochastic integral equations in the plane. We establish the existence and uniqueness of solutions to such equations as well as their additional properties.
TL;DR: In this article, the existence, uniqueness, and exponential asymptotic behavior of mild solutions to stochastic integrodifferential delay evolution equations are studied, assuming that the non-delay part generates a C0-semigroup.
Abstract: In this work, we study the existence, uniqueness, and exponential asymptotic behavior of mild solutions to stochastic integrodifferential delay evolution equations. We assume that the non-delay part generates a C0-semigroup.
TL;DR: In this article, the existence of affine realizations for Levy driven interest rate term structure models under the real-world probability measure was investigated under an assumed risk-neutral probability measure.
Abstract: We investigate the existence of affine realizations for Levy driven interest rate term structure models under the real-world probability measure, which so far has only been studied under an assumed risk-neutral probability measure. For models driven by Wiener processes, all results obtained under the risk-neutral approach concerning the existence of affine realizations are transferred to the general case. A similar result holds true for models driven by compound Poisson processes with finite jump size distributions. However, in the presence of jumps with infinite activity we obtain severe restrictions on the structure of the market price of risk; typically, it must even be constant.
TL;DR: In this article, the authors generalized Carmona and Touzi's results when the price process is allowed to jump to the context of jump diffusion processes and characterized the value function of each ordinary stopping time problem as the unique viscosity solution of the associated Hamilton-Jacobi-Bellman variational inequality.
Abstract: In their paper, Carmona and Touzi [8] studied an optimal multiple stopping time problem in a market where the price process is continuous. In this article, we generalize their results when the price process is allowed to jump. Also, we generalize the problem associated to the valuation of swing options to the context of jump diffusion processes. We relate our problem to a sequence of ordinary stopping time problems. We characterize the value function of each ordinary stopping time problem as the unique viscosity solution of the associated Hamilton–Jacobi–Bellman variational inequality.
TL;DR: In this article, the authors considered anisotropic self-similar random fields, in particular, the fractional Brownian sheet (fBs), and proved that the structure of selfsimilar Gaussian fields can be substantially more involved then the structure structure of Self-Similar Gaussian processes.
Abstract: We consider anisotropic self-similar random fields, in particular, the fractional Brownian sheet (fBs). This Gaussian field is an extension of fractional Brownian motion. It is well known that the fractional Brownian motion is a unique Gaussian self-similar process with stationary increments. The main result of this article is an example of a Gaussian self-similar field with stationary rectangular increments that is not an fBs. So we proved that the structure of self-similar Gaussian fields can be substantially more involved then the structure of self-similar Gaussian processes. In order to establish the main result, we prove some properties of covariance function for self-similar fields with rectangular increments. Also, using Lamperti transformation, we obtain properties of covariance function for the corresponding stationary fields.
TL;DR: In this article, the existence of mild solutions to a functional differential equation with delay and random effects was studied using a random fixed point theorem with a stochastic domain, and an example is included to illustrate their results.
Abstract: The authors study the existence of mild solutions to a functional differential equation with delay and random effects. They use a random fixed point theorem with a stochastic domain. An example is included to illustrate their results.
TL;DR: In this article, a Cameron-Martin type quasi-invariance theorem for subordinate Brownian motion is presented, and a gradient operator on the path space of subordinate Brownians is constructed.
Abstract: We present a Cameron–Martin type quasi-invariance theorem for subordinate Brownian motion. As applications, we establish an integration by parts formula and construct a gradient operator on the path space of subordinate Brownian motion, and obtain some canonical Dirichlet forms. These findings extend the corresponding classical results for Brownian motion.
TL;DR: In this paper, a strong approximation theorem for the local times of the weighted bootstrap process by the local time of a Brownian bridge has been proved, asymptotically.
Abstract: This article is mainly concerned with the local times of the weighted bootstrap process. We prove a strong approximation theorem for the local time of the weighted bootstrap process by the local time of a Brownian bridge. We consider also the local time of the compound empirical processes that can be seen, asymptotically, as the local time of the convolution of two independent Gaussian processes.
TL;DR: In this article, the authors introduce three types of covariance matrix structures for Gaussian or elliptically contoured vector random fields in space and/or time, which include fractional, bifractional, and trifractal vector Brownian motions as special cases.
Abstract: This article introduces three types of covariance matrix structures for Gaussian or elliptically contoured vector random fields in space and/or time, which include fractional, bifractional, and trifractional vector Brownian motions as special cases, and reveals the relationships among these vector random fields, with an orthogonal decomposition established for the multifractional vector Brownian motion.
TL;DR: In this article, the authors prove pathwise uniqueness and strong existence of solutions for stochastic reaction-diffusion systems with a locally Lipschitz continuous reaction term of polynomial growth and Holder continuous multiplicative noise.
Abstract: We prove pathwise uniqueness and strong existence of solutions for stochastic reaction-diffusion systems with a locally Lipschitz continuous reaction term of polynomial growth and Holder continuous multiplicative noise. Under additional assumptions on the coefficients, we also prove positivity of the solutions.
TL;DR: In this paper, the authors consider not only stochastic differential equations driven by the Wiener process, but also processes with stationary increments from the view points of time series analysis for mathematical finance.
Abstract: In this article, we consider not only stochastic differential equations driven by the Wiener process but also by processes with stationary increments from the view points of time series analysis for mathematical finance. Corresponding to Black-Scholes type stochastic differential equations, we consider difference equations defined by weakly dependent sequence of random vectors and examine the asymptotic behavior of their solutions.
TL;DR: In this paper, a comparison theorem for multidimensional stochastic differential equations driven by G-Brownian motion (G-SDEs) was proved for two G-diffusion processes.
Abstract: In this article, we prove a comparison theorem for multidimensional stochastic differential equations driven by G-Brownian motion (G-SDEs, for short). Moreover, we obtain respectively the sufficient conditions and necessary conditions of the monotonicity and order-preservation for two multidimensional G-diffusion processes. Finally, we give some applications.
TL;DR: In this paper, a probabilistic Feynman-Kac representation is used to transform suitable smooth functions into hard bounds for the solution to boundary value and obstacle problems for elliptic partial differential equations.
Abstract: We transform suitable smooth functions into hard bounds for the solution to boundary value and obstacle problems for elliptic partial differential equations based on the probabilistic Feynman-Kac representation. Unlike standard approximate solutions, hard solution bounds are intended to limit the location of the solution, possibly to a large extent, and, thus, have the potential to be very useful information. Our approach requires two main steps. First, the violation of sufficient conditions is quantified for the test function to be a hard bounding function. After extracting those violation terms from the Feynman-Kac representation, it remains to deal with a boundary value problem with constant input data. Although the probabilistic Feynman-Kac representation is employed, the resulting numerical method is deterministic without the need for sophisticated probabilistic numerical methods, such as sample paths generation of reflected diffusion processes. Throughout this article, we provide numerical examples ...