L′evy 过程驱动的BSDE的反比较定理与Jensen不等式

本文研究了由一维L′evy过程驱动的倒向随机微分方程（BSDE）的反比较定理。利用一般g -期望下BSDE的反比较定理的证明方法,推导出了一般f -期望下BSDE的反比较定理,并给出了一般f -期望下Jensen不等式成立的充分必要条件。     

Lévy过程驱动的BSDE的反比较定理与Jensen不等式（英文）

本文研究了由一维Lévy过程驱动的倒向随机微分方程(BSDE)的反比较定理.利用一般g-期望下BSDE的反比较定理的证明方法,推导出了一般f-期望下BSDE的反比较定理,并给出了一般f-期望下Jensen不等式成立的充分必要条件.     

与拉回度量相关泛函的f-稳态映射的刘维尔型定理

本文研究了与拉回度量相关的广义泛函Φ_f.利用f-应力能力张量的方法,得到f-稳态映射的单调公式,消灭定理以及常Dirichlet边值问题在星型区域上的唯一常值解.     

条件g-期望的矩不等式

找到了几个使条件g-期望的矩不等式在一般意义下成立的关于g和g-期望的充分条件.     

The monodromy of a series of hypersurface singularities

Let {f=0} be a hypersurface inC ⁿ⁺¹ with a 1-dimensional singular set Σ. We consider the series of hypersurfaces {f+ɛx ^N=0} wherex is a generic linear form. We derive a formula, which relates the characteristic polynomials of the monodromies off andf+ɛx ^N. Other ingredients in this formula are the horizontal and the vertical monodromies of the transversal (isolated) singularities on each branch of the singular set. We use polar curves and the carrousel method in the proof. The formula is a generalization of the Iomdin formula for the Milnor numbers: μ(f+ɛx ^N )=μ _n (f)−μ _n −1(f)+Ne ₀(Σ)     

具有势函数的弱f-稳态映射的若干结果

本文研究了与拉回度量有关广义泛函Φ_f,H.利用应力能量张量的方法,得到具有势函数的弱f-稳态映射的一些刘维尔型定理.     

含故障点的加强超立方体中路和圈的嵌入

本文研究了含故障点的n-维加强超立方体Q_n,k中的路和圈嵌入的问题.充分分析了加强超立方体网络的潜在特性,利用了构造的方法.得到了含2n-4个故障点的加强超立方体Q_n,k中含长为2ⁿ-2f的容错圈的结论,推广了折叠超立方体网络中1-点容错圈嵌入的结果.其中折叠超立方体网络为加强超立方体网络的一种特殊情况.     

有理函数域上的Gross猜想

研究分圆函数域扩张k(Λ_f)/k情形下的Gross猜想, 其中k=F_q(t)是有理函数域, f是k上的首一多项式.通过直接计算,证明了在Fermat曲线(即f=t(t−1))情形时猜想成立.当f为不可约多项式时,证明了Gross猜想和Weil互反律等价.对一般情形,证明了弱Gross猜想成立.     

一类非线性椭圆方程的刘维尔型定理

设（Mⁿ,g）是一个n维非紧的完备黎曼流行.本文考虑有正解的非线性椭圆方程△_fu+au log u=0的刘维尔型定理,其中a是一个非零常数.利用Bochner公式和极大值原理,获得了以上方程在Bakry-Emery里奇曲率有下界时正解的Li-Yau型梯度估计和某些有关的刘维尔理论,推广了文献[7]的结果.     

On arithmetical functions related to the Ramanujan sum

Let m and n be positive integers, and μ the M"bius function. And let S _f(m,n) be the function defined by , where f is an arithmetical function. We show that this function has many properties like the Ramanujan sum. Firstly we study the partial summation formula involving S _f(m,n) and taking f=μ, we obtain the Dirichlet series with the coefficients Sμ(m,n) and Sμ(m,n)d(m). Moreover we show a certain property which is analogous to the orthogonality relation of the Ramanujan sums. This revised version was published online in June 2006 with corrections to the Cover Date.     

Representation Theorem for Generators of Quadratic BSDEs

In this paper, we establish a general representation theorem for generator of backward stochastic differential equation (BSDE), whose generator has a quadratic growth in z. As some applications, we obtain a general converse comparison theorem of such quadratic BSDEs and uniqueness theorem, translation invariance for quadratic g-expectation.     

Reflected and Doubly Reflected BSDEs for Levy Processes： Solutions and Comparison

In this paper we study reflected and doubly reflected backward stochastic differential equations （BSDEs, for short） driven by Teugels martingales associated with L~vy process satisfying some moment condi- tions and by an independent Brownian motion. For BSDEs with one reflecting barrier, we obtain a comparison theorem using the Tanaka-Meyer formula. For BSDEs with two reflecting barriers, we first prove the existence and uniqueness of the solutions under the Mokobodski＇s condition by using the Snell envelope theory and then we obtain a comparison result.     

A converse comparison theorem for anticipated BSDEs and related non-linear expectations

The converse comparison theorem has received much attention in the theory of backward stochastic differential equations (BSDEs). However, no such theorem has been proved for anticipated BSDEs. In this paper, we derive a converse comparison theorem by first giving an existence and uniqueness theorem for adapted solutions of anticipated BSDEs with a stopping time and then related to (f,δ)$(f,\delta )$-expectations induced by anticipated BSDEs.     

BSDEs driven by time-changed Lévy noises and optimal control

We study backward stochastic differential equations (BSDEs) for time-changed Lévy noises when the time-change is independent of the Lévy process. We prove existence and uniqueness of the solution and we obtain an explicit formula for linear BSDEs and a comparison principle. BSDEs naturally appear in control problems. Here we prove a sufficient maximum principle for a general optimal control problem of a system driven by a time-changed Lévy noise. As an illustration we solve the mean–variance portfolio selection problem.     

A Converse Comparison Theorem for <Emphasis Type="Italic">g</Emphasis>-Expectations

With the help of a limit property of solutions of backward stochastic differential equations(BSDEs),this paper establishes a converse comparison theorem for deterministic generators g of BSDEs under the assumption g(t, y, 0) ≡0.     

Limit theorem and uniqueness theorem of backward stochastic differential equations

This paper establishes a limit theorem for solutions of backward stochastic differential equations (BSDEs). By this limit theorem, this paper proves that, under the standard assumption g(t,y,0) = 0, the generator g of a BSDE can be uniquely determined by the corresponding g-expectationεg;this paper also proves that if a filtration consistent expectation S can be represented as a g-expectationεg, then the corresponding generator g must be unique.     

Lévy-Driven Carma Processes

Properties and examples of continuous-time ARMA (CARMA) processes driven by Lévy processes are examined. By allowing Lévy processes to replace Brownian motion in the definition of a Gaussian CARMA process, we obtain a much richer class of possibly heavy-tailed continuous-time stationary processes with many potential applications in finance, where such heavy tails are frequently observed in practice. If the Lévy process has finite second moments, the correlation structure of the CARMA process is the same as that of a corresponding Gaussian CARMA process. In this paper we make use of the properties of general Lévy processes to investigate CARMA processes driven by Lévy processes {W(t)} without the restriction to finite second moments. We assume only that W (1) has finite r-th absolute moment for some strictly positive r. The processes so obtained include CARMA processes with marginal symmetric stable distributions.     

Non-linear degenerate integro-partial differential evolution equations related to geometric Lévy processes and applications to backward stochastic differential equations

We prove a comparison principle for unbounded semicontinuous viscosity sub- and supersolutions of non-linear degenerate parabolic integro-partial differential equations coming from applications in mathematical finance in which geometric Lévy processes act as the underlying stochastic processes for the assets dynamics. As a consequence of the “geometric form” of these processes, the comparison principle holds without assigning spatial boundary data. We present applications of our result to (i) backward stochastic differential equations (BSDEs) and (ii) pricing of European and American derivatives via BSDEs. Regarding (i), we extend previous results on BSDEs in a Lévy setting and the connection to semilinear integro-partial differential equations.     

Successful couplings for a class of stochastic differential equations driven by Lévy processes

By constructing proper coupling operators for the integro-differential type Markov generator, we establish the existence of a successful coupling for a class of stochastic differential equations driven by Lévy processes. Our result implies a new Liouville theorem for space-time bounded harmonic functions with respect to the underlying Markov semigroups, and it is sharp for Ornstein-Uhlenbeck processes driven by ??-stable Lévy processes.     

Dependence properties and comparison results for Lévy processes

In this paper we investigate dependence properties and comparison results for multidimensional Lévy processes. In particular we address the questions, whether or not dependence properties and orderings of the copulas of the distributions of a Lévy process can be characterized by corresponding properties of the Lévy copula, a concept which has been introduced recently in Cont and Tankov (Financial modelling with jump processes. Chapman & Hall/CRC, Boca Raton, 2004) and Kallsen and Tankov (J Multivariate Anal 97:1551–1572, 2006). It turns out that association, positive orthant dependence and positive supermodular dependence of Lévy processes can be characterized in terms of the Lévy measure as well as in terms of the Lévy copula. As far as comparisons of Lévy processes are concerned we consider the supermodular and the concordance order and characterize them by orders of the Lévy measures and by orders of the Lévy copulas, respectively. An example is given that the Lévy copula does not determine dependence concepts like multivariate total positivity of order 2 or conditionally increasing in sequence. Besides these general results we specialize our findings for subfamilies of Lévy processes. The last section contains some applications in finance and insurance like comparison statements for ruin times, ruin probabilities and option prices which extends the current literature. Anja Blatter was supported by the Deutsche Forschungsgemeinschaft (DFG).