L\'evy过程驱动的正倒向随机系统的随机最大值原理

研究了由Teugels鞅和与之独立的多维Brown运动共同驱动的正倒向随机控制系统的最优控制问题. 这里Teugels鞅是一列与L\'{e}vy 过程相关的两两强正交的正态鞅 (见Nualart, Schoutens 在2000年的结果). 在允许控制值域为一非空凸闭集假设下, 采用凸变分法和对偶技术获得了最优控制存在所满足的充分和必要条件. 作为应用, 系统研究了线性正倒向随机系统的二次最优控制问题(简记为FBLQ问题), 通过相应的随机哈密顿系统对最优控制 进行了对偶刻画. 这里的随机哈密顿系统是由Teugels鞅和多维Brown运动共同驱动的线性正倒向随机微分方程, 其由状态方程、伴随方程和最优控制的对偶表示共同来构成.     

GENERALIZED ROSENTHAL＇S INEQUALITY FOR BANACH-SPACE-VALUED MARTINGALES

A generalized Rosenthal＇s inequality for Banach-space-valued martingales is proved, which extends the corresponding results in the previous literatures and characterizes the p-uniform smoothness and q-uniform convexity of the underlying Banach space. As an application of this inequality, the strong law of large numbers for Banach-space-valued martingales is also given.     

两指标B值强鞅的大数定律

证明了两指标B值强鞅的强弱大数定律的几个重要结果,得到了Banach空间的p一致光滑特征的2个新刻划,推广和改进了单指标B值鞅和两指标B值鞅的有关结果.     

Complex valued multiparameter stochastic integrals

In this paper we will consider the properties of various stochastic integrals over a complex valued, two parameter Wiener process. Integrals of this type exhibit pleasant features which do not appear within the real framework. They are stable under approximation and viewed in relation with analytic functions, they typically satisfy an ordinary chain rule. This in turn gives rise to several nice representation formulas.     

统计递归集中的鞅性质

主要研究了统计递归集中一些相关的随机过程是否具有鞅性质,给出了一些充分条件以保证：生成统计递归集的统计压缩算子的 Ｌｉｐｓｃｈｉｔｚ 系数和所构成的随机过程具有鞅性质 这对研究统计递归集的维数及测度是有用的     

Brownian sheet images and Bessel-Riesz capacity

We show that the image of a 2-dimensional set under -dimensional, 2-parameter Brownian sheet can have positive Lebesgue measure if and only if the set in question has positive ()-dimensional Bessel-Riesz capacity. Our methods solve a problem of J.-P. Kahane.

     

Continuity of the Multifractal Spectrum of a Random Statistically Self-Similar Measure

Until now [see Kahane;⁽¹⁹⁾ Holley and Waymire;⁽¹⁶⁾ Falconer;⁽¹⁴⁾ Olsen;⁽²⁹⁾ Molchan;⁽²⁸⁾ Arbeiter and Patzschke;⁽¹⁾ and Barral⁽³⁾] one determines the multifractal spectrum of a statistically self-similar positive measure of the type introduced, in particular by Mandelbrot,^{(26, 27)} only in the following way: let be such a measure, for example on the boundary of a c-ary tree equipped with the standard ultrametric distance; for 0, denote by E the set of the points where possesses a local Hölder exponent equal to , and dim E the Hausdorff dimension of E ; then, there exists a deterministic open interval I *₊ and a function f: I *₊ such that for all in I, with probability one, dim E =f(). This statement is not completely satisfactory. Indeed, the main result in this paper is: with probability one, for all I, dim E =f(). This holds also for a new type of statistically self-similar measures deduced from a result recently obtained by Liu.⁽²²⁾ We also study another problem left open in the previous works on the subject: if =inf(I) or =sup(I), one does not know whether E is empty or not. Under suitable assumptions, we show that E ø and calculate dim E .     

Exact inequalities for sums of asymmetric random variables, with applications

Let be independent identically distributed random variables each having the standardized Bernoulli distribution with parameter . Let if and . Let . Let f be such a function that f and f′′ are nondecreasing and convex. Then it is proved that for all nonnegative numbers one has the inequality where . The lower bound on m is exact for each . Moreover, is Schur-concave in . A number of corollaries are obtained, including upper bounds on generalized moments and tail probabilities of (super)martingales with differences of bounded asymmetry, and also upper bounds on the maximal function of such (super)martingales. Applications to generalized self-normalized sums and t-statistics are given.      

带利率的特殊双险种风险模型的破产概率

研究了保费为一复合随机过程且含利率因素的特殊双险种风险模型,给出了此模型下保险公司稳定经营的必要条件;证明了调节系数的存在性;用鞅方法讨论了此模型的破产概率上界.