一类1 < α < 2非局部条件下的脉冲分数阶微分方程mild解的存在性

本文研究了一类1 < α < 2非局部条件下的脉冲分数阶偏微分方程mild解的存在性问题.利用解算子的相关性质及Krasnoselskii不动点理论的方法,获得了这类方程的mild解并予以证明,且得到了解的存在性结果.     

Tropical differential equations

Tropical differential equations are introduced and an algorithm is designed which tests solvability of a system of tropical linear differential equations within the complexity polynomial in the size of the system and in the absolute values of its coefficients. Moreover, we show that there exists a minimal solution, and the algorithm constructs it (in case of solvability). This extends a similar complexity bound established for tropical linear systems. In case of tropical linear differential systems in one variable a polynomial complexity algorithm for testing its solvability is designed.We prove also that the problem of solvability of a system of tropical non-linear differential equations in one variable is NP-hard, and this problem for arbitrary number of variables belongs to NP. Similar to tropical algebraic equations, a tropical differential equation expresses the (necessary) condition on the dominant term in the issue of solvability of a differential equation in power series.     

SDEs with constraints driven by semimartingales and processes with bounded p-variation

We study the existence, uniqueness and stability of solutions of general stochastic differential equations with constraints driven by semimartingales and processes with bounded $p$-variation. Applications to SDEs with constraints driven by fractional Brownian motion and standard Brownian motion are given.     

Lp solutions of backward stochastic differential equations with jumps

Given $p\in (1,2)$, we study ${\mathbb{L}}^p$ solutions of a multi-dimensional backward stochastic differential equation with jumps (BSDEJ) whose generator may not be Lipschitz continuous in $(y,z)$-variables. We show that such a BSDEJ with $p$-integrable terminal data admits a unique ${\mathbb{L}}^p$ solution by approximating the monotonic generator by a sequence of Lipschitz generators via convolution with mollifiers and using a stability result.     

Coupling and exponential ergodicity for stochastic differential equations driven by Lévy processes

We present a novel idea for a coupling of solutions of stochastic differential equations driven by Lévy noise, inspired by some results from the optimal transportation theory. Then we use this coupling to obtain exponential contractivity of the semigroups associated with these solutions with respect to an appropriately chosen Kantorovich distance. As a corollary, we obtain exponential convergence rates in the total variation and standard $L^1$-Wasserstein distances.     

面向信息化战争的广义兰切斯特战争模型

21世纪是一个信息化时代,为了更好的面向信息化战争.通过对传统的兰切斯特微分方程进行改进,考虑了信息化的诸多因素,引入了战场情报感知系数和信息优势系数.构建了数学模型,并对其进行差分处理求解,比较分析了交战双方对战场感知系数变化和信息优势系数变化情况下,对战争的影响.以海湾战争为例,检验了模型的可行性.     

非齐次树上m重非齐次马氏链的一类强偏差定理

研究给出了非齐次树上m重非齐次马氏链的一类强偏差定理.     

分数扩散模型的参数估计及其应用

研究分数扩散模型的参数估计及其应用问题.分数扩散模型是一类由分数Brownian运动驱动的随机微分方程.主要结果有:(1)利用二次变差方法给出模型中扩散系数的估计量,通过最小二乘法给出模型中漂移系数的估计量;(2)证明这些估计量的一致收敛性和渐近正态性;(3)利用MCMC方法对此估计量进行验证,并通过R软件将上述模型以及参数估计量应用到SHIBOR利率中进行实证研究.     

基于随机微分博弈的动态投入产出问题研究

基于W Leontief提出的动态投入产出模型,讨论了考虑随机因素的离散时间下的最优策略设计问题.利用博弈论的思想,把动态投入产出系统抽象为离散时间的博弈模型,运用鞍点均衡策略设计出求解该投入产出问题的新方法,为宏观经济决策提供依据.     

粘性解框架下的完全耦合正倒向随机系统最优控制问题的验证定理

研究了完全耦合正倒向随机控制系统的最优控制问题.得到了粘性解框架下的,控制变量同时出现在正倒向随机系统的漂移项和扩散项中的最优控制问题的验证定理.还讨论了验证定理在构造随机最优反馈控制中的应用.