Set-Valued Stochastic Integrals and Equations with Respect to Two-Parameter Martingales

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.     

Stochastic Integrals of Continuous Local Martingales,I

Consider a continuous local martingale X. We say that X satisfies the representation property if any martingale Y of X can be represented as stochastic ITǒ integral of X. Using the method of random time change systematically, in the present paper the representation problem for continuous local martingales is treated. We describe a class of martingales Y that can be represented as stochastic integral of X by probabilistic conditions. This leads to sufficient conditions for the representation property of X being true. Besides, an interesting characterization of continuous processes with independent increments is obtained. In part II. we proceed with general examples, applications to the n-dimensional case, and, in particular, to the n-dimensional time change of continuous local martingales with orthogonal components.     

Stochastic Integrals of Continuous Local Martingales,II

Consider a continuous local martingale X. We say that X satisfies the representation property if any martingale Y of X can be represented as stochastic ITÔ integral of X. On the basis of part I of the present paper, in section 4 several general examples of continuous local martingales X satisfying the representation property are given: Stochastic continuous GAUSSian martingales, processes with conditionally independent increments, stopped continuous local martingales, random time change of WIENER processes, weak solutions of stochastic differential equations. Theorem 7 states that every (homogeneous) continuous strong MARKOV local martingale has the representation property. In section 5, the results of part I are applied to n-dimensional continuous local martingales and analogous representation results are obtained. In section 6, we consider an application of section 5 to the n-dimensional time change for reducing every n-dimensional continuous local martingale with orthogonal components to the WIENER process. This improves a theorem of F. B. KNIGHT and simplifies its proof considerably.     

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.     

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.      

On Convergence to Stochastic Integrals

Weak convergence of various general functionals of partial sums of dependent random variables to stochastic integrals now plays a major role in modern statistics theory. In this paper, we obtain the weak convergence of various general functionals of partial sums of causal process by means of the method which was introduced in Jacod and Shiryaev (Limit theorems for stochastic processes. Springer, Berlin, 2003).     

On the Strong Law of Large Numbers for Multivariate Martingales with Continuous Time

We prove the strong law of large numbers for vector martingales with arbitrary operator normalizations. From the theorem proved, we deduce several known results on the strong law of large numbers for martingales with continuous time.     

有界可料过程关于集值平方可积鞅的集值随机积分

本文定义了一类有界可料过程关于集值平方可积鞅的集值随机积分,并研究了集植随机积分的性质。此为建立集值随机分析的理论奠定了基础。     

定积分计算的参数变换法

两种围道得到同一个含参数复函数定积分的表达式,一个表达式是带参数三角函数的定积分,另一个表达式是带参数的特殊函数.对参数进行积分和级数展开等操作,对比参数展开系数,得到了一些三角函数定积分的值.     

A Heisenberg Inequality for Stochastic Integrals

Using the products of exponential functions, a family of unitary operators will be defined for all square integrable continuous martingale processes having a deterministic quadratic variation. A Heisenberg inequality for stochastic integrals will be proved using this family of operators.     

Strong Markov Continuous Local Martingales and Solutions of One-Dimensional Stochastic Differential Equations (Part III)

This last part of the present paper is devoted to one-dimensional stochastic differential equations driven by a WIENER process. In Section 4, we give a survey on existence, uniqueness, and various other aspects of solutions. In Section 5, which was the starting point of the present paper, we describe the connection between strong MARKOV continuous local martingales and solutions of one-dimensional stochastic differential equations without drift.     

Nonlinear Stochastic Difference Equations Driven by Martingales

Abstract

We provide in this paper a systematic development of nonlinear stochastic difference equations driven by martingales (that depend on a spatial parameter); three such equations are considered. We begin with the existence and uniqueness of solutions and continue with the study of stochastic properties, such as the martingale and Markov properties, along with ? irreducibility and recurrence. We discuss in the final section the discrete-time flow and asymptotic flow properties of the solution process.     

Itô Formula for Free Stochastic Integrals

The objects under investigation are the stochastic integrals with respect to free Lévy processes. We define such integrals for square-integrable integrands, as well as for a certain general class of bounded integrands. Using the product form of the Itô formula, we prove the full functional Itô formula in this context.     

A Large Deviation Principle for Stochastic Integrals

Assuming that {(X _n ,Y _n )} satisfies the large deviation principle with good rate function I ^♯ , conditions are given under which the sequence of triples {(X _n ,Y _n ,X _n ⋅Y _n )} satisfies the large deviation principle. An ε-approximation to the stochastic integral is proven to be almost compact. As is well known from the contraction principle, we can derive the large deviation principle when applying continuous functions to sequences that satisfy the large deviation principle; the method showed here skips the contraction principle, uses almost compactness and can be used to derive a generalization of the work of Dembo and Zeitouni on exponential approximations. An application of the main result to stochastic differential equations is given, namely, a Freidlin-Wentzell theorem is obtained for a sequence of solutions of SDE’s.     

On One Sufficient Condition for the Validity of the Strong Law of Large Numbers for Martingales

We prove a theorem on the strong law of large numbers for martingales. The existence of higher moments is not assumed. From the theorem proved, we deduce numerous well-known results on the strong law of large numbers both for martingales and for sequences of sums of independent random variables.     

Graph-Theoretic Approach to Stochastic Integrals with Clifford Algebras

Given a fixed probability space (Ω,ℱ,ℙ) and m≥1, let X(t) be an L²(Ω) process satisfying necessary regularity conditions for existence of the mth iterated stochastic integral. For real-valued processes, these existence conditions are known from the work of D. Engel. Engel’s work is extended here to L²(Ω) processes defined on Clifford algebras of arbitrary signature (p,q), which reduce to the real case when p=q=0. These include as special cases processes on the complex numbers, quaternion algebra, finite fermion algebras, fermion Fock spaces, space-time algebra, the algebra of physical space, and the hypercube. Next, a graph-theoretic approach to stochastic integrals is developed in which the mth iterated stochastic integral corresponds to the limit in mean of a collection of weighted closed m-step walks on a growing sequence of graphs. Combinatorial properties of the Clifford geometric product are then used to create adjacency matrices for these graphs in which the appropriate weighted walks are recovered naturally from traces of matrix powers. Given real-valued L²(Ω) processes, Hermite and Poisson-Charlier polynomials are recovered in this manner.     

随机积分的拟必然逼近

该文证明了随机积分的拟必然逼近. 基于此,作者用更简化的方法得到并拓展了文献[8]的主要结果. 并且, 该方法可以应用到光滑两参数鞅情形.     

Stochastic Integrals with Respect to Consistent Random Measures

We consider integrals of random mappings with respect to consistent random measures in C([0; 1]).