On the pathwise uniqueness of solutions of stochastic differential equations driven by symmetric stable Lévy processes1

We investigate a sufficient condition for pathwise uniqueness property for 1D stochastic differential equation driven by symmetric α-stable Lévy process, where α ∈ (1, 2).     

Approximation theorems for set-valued stochastic integrals

The article is devoted to new properties of Aumann, Lebesgue, and Itô set-valued stochastic integrals considered in papers [1 Kisielewicz, M. (2014). Properties of generalized set-valued stochastic integrals. Discuss. Math. (DICO) 34:131–147. [Google Scholar],2 Kisielewicz, M., Michta, M. (2017). Integrably bounded set-valued stochastic integrals. J. Math. Anal. Appl. 449:1893–1910.[Crossref], [Web of Science ®] , [Google Scholar]]. In particular, it contains some approximation theorems for Aumann and Itô set-valued stochastic integrals. Hence, in particular, it follows that Aumann and Lebesgue set-valued stochastic integrals cover a.s., both for measurable and IF-nonanticipative integrably bounded set-valued stochastic processes.     

On Maximal Inequalities for Stable Stochastic Integrals

Sharp maximal inequalities in large and small range are derived for stable stochastic integrals. In order to control the tail of a stable process, we introduce a truncation level in the support of its Lévy measure: we show that the contribution of the compound Poisson stochastic integral is negligible as the truncation level is large, so that the study is reduced to establish maximal inequalities for the martingale part with a suitable choice of truncation level. The main problem addressed in this paper is to give upper bounds which remain bounded as the parameter of stability of the underlying stable process goes to 2. Applications to estimates of first passage times of symmetric stable processes above positive continuous curves complete this work.      

Absolute Continuity of Joint Laws of Multiple Stable Stochastic Integrals

We are interested in the laws of multiple stable stochastic integrals defined by LePage series representation in references(3,10,11). We continue the study started in Ref. 3 and give conditions ensuring absolute continuity of joint laws of stable integrals. To this end, we apply a stratification method on the Skorohod space on which we first take back the problem.     

On Set-Valued Stochastic Integrals

We define a set-valued stochastic integral with respect to a 1-dimensional Brownian motion. The paper develops multivalued analogs to the theory of singlevalued stochastic integrals. It is expected that these results will be useful to study set-valued and fuzzy stochastic analysis.     

Two-Parameter Fuzzy-Valued Stochastic Integrals and Equations

This article is concerned with notions of fuzzy-valued stochastic integrals driven by two-parameter martingales and increasing processes. We present their main properties and formulate next two-parameter fuzzy-valued stochastic integral equations. We establish the existence and uniqueness of solutions to such equations as well as their additional properties.     

Large Deviations for Stochastic Volterra Equations in the Plane

We prove a Large Deviation Principle for the family of solutions of Volterra equations in the plane obtained by perturbation of the driving white noise. One of the motivations for the study of such class of equations is provided by non-linear hyperbolic stochastic partial differential equations appearing in the construction of some path-valued processes on manifolds. The proof uses the method developped by Azencott for diffusion processes. The main ingredients are exponential inequalities for different classes of two-parameter stochastic integrals; these integrals are related to the representation of the stochastic term in the differential equation as a representable semimatringale.     

DISCRETIZATION OF JUMP STOCHASTIC DIFFERENTIAL EQUATIONS IN TERMS OF MULTIPLE STOCHASTIC INTEGRALS

1.IntroductionFOrthestrongdiscretizationofSDEs,anynumericalmethodwhichonlydependsonthevaluesofBrownianpathsorPoissonpathsatthepartitionnodescannotachieveanorderhigherthan0.5ingeneral[')'1'].Thereforetheevaluationofmultiplestochasticintegralsontheintervalsbetweennodesisamajorobstaclethatmustbeovercome.Someattemptshavebeenmadepreviouslyindifferentapproachestoapproximatemul-tiplestochasticintegrals.[2]suggestsanapproximationintermsofFourierGaussiancoefficientsoftheBrownianbridgeprocess.Asthel…     

关于随机积分的一点注记

本文给出随机积分的一种新的逼近方法．构造了一种统一而具体的构造程序，并利用这一程序解决了有关随机积分的分布和随机微分方程的变量代换的问题．     

Stochastic Theta Method for a Reflected Stochastic Differential Equation

In this article, a stochastic theta method for a reflected stochastic differential equation is proposed. When the parameter θ = 0, this method coincides with the projection Euler scheme; while when the parameter θ = 1, it is called an implicit projection Euler scheme which is first proposed in this article. Under some conditions, the strong convergence and the A-stability of this numerical scheme are proved.     

An Isometric Approach to Generalized Stochastic Integrals

The possibility to extend the classical Ito's construction of stochastic integrals is studied. This construction can be applied to fractional Brownian motions with Hurst index H(0, 1/2). A change of variables formula for fractional Brownian motions in terms of the stochastic integrals is given.     

Martingale representation theorem for set-valued martingales

The present paper contains a martingale representation theorem for set-valued martingales defined on a filtered probability space with a filtration generated by a Brownian motion. It is proved that such type martingales can be defined by some generalized set-valued stochastic integrals with respect to a given Brownian motion. The main result of the paper is preceded by short part devoted to the definition and some properties of generalized set-valued stochastic integrals.     

Characterization of Multidimensional Stable Random Measures by Means of Vector Measures

Abstract

In connection with a symmetric α stable random measure Φ on a measurable space (F, ?) with values in R ^d , a complete metric space of symmetric finite measures on S _d?1 is constructed, and is employed to characterize the law of Φ by a unique positive measure on ? and a unique function on F × R ^d . The stochastic integral ∫ _F f d Φ is also defined for certain d × d matrix valued functions f, which for α = 2 reduces to the Wiener–Masani integral.     

分数Brown 运动驱动的非 Lipschitz随机微分方程

讨论了一类带分数Brown 运动的非Lipschitz 增长的随机微分方程适应解的存在唯一性。关于分数 Brown 运动的随机积分有多种定义,本文使用一种广义 Stieltjes积分定义方法,利用这种积分的性质,建立了一类由标准 Brown 运动和一个 Hurst 指数H ∈(1/2,1)的分数Brown 运动共同驱动的、系数为非Lipschitz 增长的随机微分方程适应解的存在唯一性定理。     

分数布朗单的幂函数随机积分逼近

本文研究了分数布朗单的逼近问题.利用Wiener积分,得到了分数布朗单的幂函数型随机积分逼近.     

Strong solution of Itô type set-valued stochastic differential equation

In this paper, we shall firstly illustrate why we should introduce an It5 type set-valued stochastic differential equation and why we should notice the almost everywhere problem. Secondly we shall give a clear definition of Aumann type Lebesgue integral and prove the measurability of the Lebesgue integral of set-valued stochastic processes with respect to time t. Then we shall present some new properties, especially prove an important inequality of set-valued Lebesgue integrals. Finally we shall prove the existence and the uniqueness of a strong solution to the It5 type set-valued stochastic differential equation.     

Frechet-valued martingales and stochastic integrals

Stochastic processes with values in a separable Frechet space whose a itinuous linear functional are real-valued square integrable martingales are investigated. The coordinate measures on the Fréchet space are obtained from cylinder set measures on a Hilbert space that is dense in the Fréchet space. Real-valued stochastic integrals are defined from the Fréchet-valued martingales using integrands from the topological dual of the aforementioned Hilbert space. An increasing process with values in the self adjoint operators on the Hilbert space plays a fundamental role in the definition of stochastic integrals. For Banach-valued Brownian motion the change of variables formula of K. Itô is generalized. A converse to the construction of the measures on the Fréchet space from cylinder set measures on a Hilbert space is also obtained.     

Stochastic integrals of empirical-type processes with applications to censored regression

Motivated by the analysis of linear rank estimators and the Buckley-James nonparametric EM estimator in censored regression models, we study herein the asymptotic properties of stochastic integrals of certain two-parameter empirical processes. Applications of these results on empirical processes and their stochastic integrals to the asymptotic analysis of censored regression estimators are also given.     

Stochastic bilinear spectral systems †

A class of bilinear stochastic partial differential equations is investigated using a semigroup approach. Existence of a mild solution is obtained by proving a maximal inequality for stochastic convolution integrals with a stochastic evolution operator U(t,s) as integrand; moreover, we show the existence of a regular version in t. Under an additional assumption we show the existence of a continuous version of U (.,.) in the space of bounded operators on the state space. Finally, we analyse a p.d.e. model of a simply supported beam to illustrate the applicability of our results to modelling uncertainty in large flexible space structures     

Stochastic integral with respect to set-valued square integrable martingales

In this paper, we shall firstly illustrate why we should consider integral of a stochastic process with respect to a set-valued square integrable martingale. Secondly, we shall prove the representation theorem of set-valued square integrable martingale. Thirdly, we shall give the definition of stochastic integral of a stochastic process with respect to a set-valued square integrable martingale and the representation theorem of this kind of integrals. Finally, we shall prove that the stochastic integral is a set-valued sub-martingale.