倒向随机微分方程的Malliavin微分和共单调定理(英)

本文利用Malliavin微分的理论研究了倒向随机微分方程的解(y,z),首先利用y的Malliavin微分得到了一种比较z的方法,然后利用该方法得到了含有随机生成元的倒向随机微分方程的共单调定理.     

倒向随机微分方程的Malliavin微分和共单调定理

本文利用Malliavin微分的理论研究了倒向随机微分方程的解$(y,z)$, 首先利用$y$的Malliavin微分得到了一种比较$z$的方法, 然后利用该方法得到了含有随机生成元的倒向随机微分方程的共单调定理.     

关于带跳随机微分方程解的密度光滑性

本文利用Malliavin随机变分来讨论一类带跳随机微分方程解的密度光滑性。在退化情形并且当跳部分为零时,所给的条件就是通常的H(?)rmander条件。     

由L\'{e}vy过程驱动的仿射方程关联的无限时区的最优二次控制

讨论线性二次最优控制问题, 其随机系统是由 L\'{e}vy 过程驱动的具有随机系数而且还具有仿射项的线性随机微分方程. 伴随方程具有无界系数, 其可解性不是显然的. 利用 $\mathscr{B}\mathscr{M}\mathscr{O}$ 鞅理论, 证明伴随方程在有限 时区解的存在唯一性. 在稳定性条件下, 无限时区的倒向随机 Riccati 微分方程和伴随倒向随机方程的解的存在性是通过对应有限 时区的方程的解来逼近的. 利用这些解能够合成最优控制.     

正倒向随机微分方程解的比较定理

讨论了正倒向随机微分方程解的比较问题.阐述了正倒向随机微分方程在随机最优控制、现代金融理论中的广泛而深刻的应用, 对于一类正倒向随机微分方程, 利用Ito公式、停时等随机分析方法,通过构造辅助正倒向随机微分方程,得到了正倒向随机微分方程解的比较定理.     

带有确定生成元的倒向随机微分方程的共单调定理

对倒向随机微分方程(简记BsDE)的解(y,z),利用Malliavin微分的方法进行了研究.给出了某些关于比较z的方法,在此基础上继续研究(y,z)的某些重要性质,同时推广了Chen Zengjing等人文章中相应的结论.     

倒向随机微分方程解的Malliavin微分

讨论倒向随机微分方程Ｙｔ＝ζ＋∫＾Ｔｔｇ（ｓ,Ｙｓ,Ｚｓ）ｄｓ－∫＾ＴｔＺｓｄＷｓ解在Ｍａｌｌｉａｖｉｎ微分意义下的可微性,并得到其Ｍａｌｌｉａｖｉｎ二阶微分仍然满足一个倒向随机微分方程。用迭代方法构造一个随机序列（Ｙ＾ｎ．Ｚ＾ｎ．）,证明在Ｍａｌｌｉａｖｉｎ微分意义下二阶可微,同时证明了它在Ｓｏｂｏｌｅｖ空间Ｄ２,２则中收敛于一个线性倒向随机微分方程的解。     

正倒向随机微分方程的适应解及其比较定理(英文)

研究了一类正倒向随机微分方程的适应解 ,其中正向方程不需要满足非退化条件 .我们证明了在某些单调条件下 ,正倒向随机微分方程存在唯一的适应解 ,并给出了该正倒向随机微分方程的比较定理 .     

非Lipschitz条件下的包含下微分算子的带跳倒向随机微分方程(英文)

本文在非Lipschitz系数下,考虑了一类多值的倒向随机微分方程.利用极大单调算子的Yosida估计和倒向随机微分方程在非Lipschitz条件下解的存在唯一性,获得了多值带跳的倒向随机微分方存在唯一解的结论.     

倒向随机Volterra积分方程适应解的表示

倒向随机Volterra积分方程可以看作(确定性)Volterra积分方程和倒向随机微分方程的推广,在随机最优控制理论和数学金融学中有诸多应用.本文利用正倒向随机微分方程适应解表示的思想,得到所研究的一类倒向随机Volterra积分方程适应解的表示.这样的结果对研究适应解的正则性以及数值计算有重要的意义.     

Higher order differentiability of solutions to backward stochastic differential equations

In this paper, we consider the differentiability in the sense of the Malliavin calculus of solutions to backward stochastic differential equations (BSDEs for short). It is known that a solution is differentiable in the sense of the Malliavin calculus and the derivative is also a solution to a linear BSDE. Under additional conditions, we will show that the higher order differentiability of a solution to a BSDE and that it also becomes a solution to a linear BSDE.     

Fractional Smoothness of Some Stochastic Integrals

We study the fractional smoothness in the sense of Malliavin calculus of stochastic integrals of the form ∫0^1Ф（Xs）dXs, where Xs is a semimartingale and Ф belongs to some fractional Sobolev space over R.     

Path regularity and explicit convergence rate for BSDE with truncated quadratic growth

We consider backward stochastic differential equations with drivers of quadratic growth (qgBSDE). We prove several statements concerning path regularity and stochastic smoothness of the solution processes of the qgBSDE, in particular we prove an extension of Zhang’s path regularity theorem to the quadratic growth setting. We give explicit convergence rates for the difference between the solution of a qgBSDE and its truncation, filling an important gap in numerics for qgBSDE. We give an alternative proof of second order Malliavin differentiability for BSDE with drivers that are Lipschitz continuous (and differentiable), and then derive an analogous result for qgBSDE.     

Γ-Convergence of some super quadratic functionals with singular weights

We study the Γ-convergence of the following functional (p > 2)

Solving the hammerstein integral equation in the irregular case by successive approximations

The branches of a solution of the nonlinear integral equation

Existence of solutions of abstract volterra equations in a banach space and its subsets

We consider a criterion and sufficient conditions for the existence of a solution of the equation

An Osgood type regularity criterion for the liquid crystal flows

In this paper, we prove an Osgood type regularity criterion for the model of liquid crystals, which says that the condition $$\sup_{2 \leq q< \infty} \int \nolimits_0^T \frac{\| \bar{S}_{q} \nabla {\bf u}(t)\|_{L^\infty}}{q \, {\rm \ln} \, q} {\rm d} t<\infty$$ implies the smoothness of the solution. Here, ${{\bar S_q=\sum\nolimits_{k=-q}^q \dot {\triangle}_k}}$ with ${\dot{\triangle}_k}$ being the frequency localization operator.     

拟线性抛物方程的弱解

该文给出了拟线性退化抛物方程pa_t{u}+pa_x{f(u)}=pa_xx{A(u(x,t))}∈R^2_+×(0,+∞) ,u(x,0)=u_0(x),x∈R 一种弱解的新定义, 利用Div Curl引理证明了解的存在性．     

Increasing functions and inverse Santaló inequality for unconditional functions

Let be a convex function and be its Legendre tranform. It is proved that if is invariant by changes of signs, then . This is a functional version of the inverse Santaló inequality for unconditional convex bodies due to J. Saint Raymond. The proof involves a general result on increasing functions on together with a functional form of Lozanovskii’s lemma. In the last section, we prove that for some c > 0, one has always . This generalizes a result of B. Klartag and V. Milman.      

Gibbs measures associated to the integrals of motion of the periodic derivative nonlinear Schrödinger equation

We study the one-dimensional periodic derivative nonlinear Schrödinger equation. This is known to be a completely integrable system, in the sense that there is an infinite sequence of formal integrals of motion \({\textstyle \int }h_k\), \(k\in {\mathbb {Z}}_{+}\). In each \({\textstyle \int }h_{2k}\) the term with the highest regularity involves the Sobolev norm \(\dot{H}^{k}({\mathbb {T}})\) of the solution of the DNLS equation. We show that a functional measure on \(L^2({\mathbb {T}})\), absolutely continuous w.r.t. the Gaussian measure with covariance \(({\mathbb {I}}+(-\varDelta )^{k})^{-1}\), is associated to each integral of motion \({\textstyle \int }h_{2k}\), \(k\ge 1\).