平方可积鞅

讨论了集值平方可积鞅和实值平方可积鞅的性质.它对集值随机过程的进一步研究将起到很重要的作用.     

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。     

局部平方可积鞅的比值不等式

本对局部平方可积鞅建立了几个比值不等式，推广了连续鞅的相应结果．     

关于局部平方可积鞅的重对数律

设(M_t,f_t,t≥0)为(Ω,f,P)上的局部平方可积鞅,(简记M ∈m_100~2),M_=0,{f_t}满足通常条件,f_0={Φ,Ω},为M~2的可料补偿,本文证明了如下结论: ⅰ)若存在可料过程k_t t≥0,k_t=0 a.s.,使得 |△M_t|≤k_t·_t~(1/2)/[2lg_2 k~2V_t)]~(1/2) a.s.,_∞=∞, 则 M_t/[2_t lg_2(e~2V_t)]~(1/2)=1 a.s. ⅱ)若存在可料过程K_t,t≥0和常数0_t~(1/2)/[21g_2(e~2V_t)]~(1/2) a.s. 则存在0<8(K)<1,↓8(K)=0,使在_∞=∞上, M_t/[2_t lg_2(e~2V_t)]~(1/2)≤1 8(K) a.s.     

局部平方可积鞅的比值不等式

本文对局部平方可积鞅建立了几个比值不等式,推广了连续鞅的相应结果.     

关于模糊集值平方可积鞅的随机积分

给出了模糊集值平方可积鞅的定义以及简单实值可料过程关于模糊集值平方可积鞅的随机积分的定义;证明了该积分仍具有模糊集值平方可积鞅的性质。     

局部平方可积鞅Chung重对数律的下界

设Ｘ＝（Ｘｔ，ｔ≥０）为零初值的局部平方可积鞅〈Ｘ，Ｘ〉＝（〈Ｘ，Ｘ〉ｔ，ｔ≥０）为具可料二阶交差，在类似于Ｋｏｌｍｏｎｏｒｏｖ最初给出的条件下，证明了局部平方可积鞅的Ｃｈｕｎｇ重对数律的下界成立，即     

局部平方可积鞅的Chug重对数律

设Ｘ＝（Ｘｔ，ｔ≥０）为局部平方可积鞅，且Ｘ０＝０〈Ｘ，Ｘ〉ｔ为其二阶可料变差。利用继续半鞅的强逼近结果，我们证明了在较弱的条件下，Ｘ的Ｃｈｕｎｇ重对数律成立，即ｐ（＾ｌｉｍｉｎｆ ｔ→∞ ＾ｓｕｐ│Ｘｓ│ ｏ≤ｓ≤ｔ／（〈ｘ，ｘ〉ｔ／ｌｏｇｌｏｇ〈Ｘ，Ｘ〉 ｔ）＾１／２＝π／根号８）＝１。     

局部平方可积鞅的Chung重对数律(英文)

设X＝（Xt，t0）为局部平方可积鞅，且Xo＝0，＜X，X＞t为其二阶可料变差．利用连续半鞅的强逼近结果，我们证明了在较弱的条件下，X的Chung重对数律成立，即     

集值拟鞅和集值一致渐近鞅

本文讨论了集值拟鞅和集值一致渐近鞅,证明了集值拟鞅与集值一致渐近鞅的选样定理,对于集值一致渐近鞅得到了一些收敛性结果,并由此刻化了空间的 Radon-Nikodym性质.     

集值L1极限鞅导出的集值测度

讨论集值L1极限鞅的一些性质,在此基础上,研究集值L1极限鞅导出的集值测度及其性质.     

集值逆鞅的收敛性

研究集值逆鞅(集值逆上鞅)在Kuratowski收敛意义,Kuratowski-Mosco收敛意义及弱收敛意义下的收敛定理.     

集值测度的Radon-Nikodym定理及集值算子的Pettis-Aumann积分表示

本文建立了 Ｂａｎａｃｈ空间集值测度的 Ｒａｄｏｎ－Ｎｉｋｏｄｙｍ定理,并给出了两类集值算子的Ｐｅｔｔｉｓ－Ａｕｍａｎｎ积分表示．     

Martingale solutions and Markov selections for stochastic partial differential equations

We present a general framework for solving stochastic porous medium equations and stochastic Navier–Stokes equations in the sense of martingale solutions. Following Krylov [N.V. Krylov, The selection of a Markov process from a Markov system of processes, and the construction of quasidiffusion processes, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973) 691–708] and Flandoli–Romito [F. Flandoli, N. Romito, Markov selections for the 3D stochastic Navier–Stokes equations, Probab. Theory Related Fields 140 (2008) 407–458], we also study the existence of Markov selections for stochastic evolution equations in the absence of uniqueness.     

Martingale solutions of a stochastic wave equation with reflection

We discuss an initial boundary value problem for a one-dimensional stochastic wave equation with reflection. For stochastic parabolic equations with reflection, there are some well-known results. However, there seems to be no existence result for a stochastic wave equation with reflection. Even for a deterministic wave equation, the problem has not been completely resolved. Our goal is to establish the existence of a martingale solution for this problem.     

一类以鞅为驱动的随机泛函微分方程强解的存在性与唯一性

在积分型Lipschitz条件下,证明了一类以连续鞅为驱动的随机泛函微分方程解的存在性与唯一性.     

Weighted Vector Valued Inequalities for Generalized Singular Integral Operators on Spaces of Homogeneous Type

ＷｅｉｇｈｔｅｄＶｅｃｔｏｒＶａｌｕｅｄＩｎｅｑｕａｌｉｔｉｅｓｆｏｒＧｅｎｅｒａｌｉｚｅｄＳｉｎｇｕｌａｒＩｎｔｅｇｒａｌＯｐｅｒａｔｏｒｓｏｎＳｐａｃｅｓｏｆＨｏｍｏｇｅｎｅｏｕｓＴｙｐｅＬｉｕｉｎｎｚｈｅ（刘岚喆）（ＭａｔｈｅｍａｔｉｃｓＤｅｐ...     

Explicit Martingale Representations for Brownian Functionals and Applications to Option Hedging

Abstract

Using Clark-Ocone formula, explicit martingale representations for path-dependent Brownian functionals are computed. As direct consequences, explicit martingale representations of the extrema of geometric Brownian motion and explicit hedging portfolios of path-dependent options are obtained.     

About the asymptotic behaviour of continuous vector-valued local martingales and application in multiple linear regression models

We study the asymptotic behaviour of the process d M ¢ m 1 M , where M is an R n -continuous vector local martingale and d M ¢ m 1 is the inverse of its predictable quadratic variation (a matrix-valued process). We also give an application (strong consistency and rate of convergence) to multiple linear regression.