Multidimensional Stochastic Analysis

多维随机分析

• 批准号: 0244737
• 负责人: Richard Bass
• 金额: --
• 依托单位: University of Connecticut
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-06-01 至 2007-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0244737&HistoricalAwards=false
• 关键词: Multidimensional; Stochastic; Analysis

0244737Bass The principal investigator will be working on problems in two areas of probability. The first is concerned with Harnack inequalities. A Harnack inequality asserts that nonnegative solutions to a partial differential equation satisfy certain boundedness estimates at points, and thus allow one to obtain pointwise estimates from global information. They are used in partial differential equations to obtain estimates on heat kernels and to prove regularity properties of solutions. They are used in probability to obtain transition density estimates and regularity properties of certain stochastic processes. The principal investigator will investigate when one can obtain Harnack inequalities for functions related to non-local operators. The operators in question have an integral term and correspond to processes with jumps. The second area of research concerns uniqueness for the solutions of stochastic differential equations arising from population models in mathematical biology. These equations describe the limit of branching diffusion processes as the number of particles increases, the mass of each particle decreases, and the branching rate increases. The branching rate and the diffusion mechanism for a particle are allowed to depend on all other particles in the system. Branching diffusions are used as models of population dynamics for a large variety of species. The equations that result are typically either infinite dimensional, degenerate, or both. The principal investigator will continue his work on proving uniqueness for these equations. It has been known for a long time that many systems in the physical and biological sciences can be modeled by stochastic processes. More recently it has been discovered that many financial and economic systems can also be so modeled. To investigate more complex systems, new types of random processes have arisen. To give an example, stock prices are often viewed as depending on a continuous random process, Brownian motion. Yet the fluctuations of stock prices often have sudden jumps, resulting from wars, new discoveries, etc. Thus it is essential to also study stochastic processes with jumps. When studying population models, one expects that the behavior of the population will be qualitatively different depending on whether the population is large or whether it is small. The research of the principal investigator is primarily concerned with two types of stochastic processes, ones with jumps, as in the stock market example, and ones concerning systems that can degenerate, as in the population example. Some of the questions that are being investigated are whether there is only one solution to the equation and whether the solution has sufficient regularity to be useful in providing new information for the model.

0244737Bass首席研究员将研究两个概率领域的问题。第一个是Harnack不等式。一个Harnack不等式断言偏微分方程的非负解在点上满足一定的有界性估计，从而允许人们从全局信息中获得逐点估计。它们用于偏微分方程中以获得热核的估计并证明解的正则性。它们在概率中用于获得某些随机过程的转移密度估计和正则性。主要研究者将调查何时可以获得Harnack不等式的功能有关的非本地运营商。所讨论的运算符有一个积分项，并且对应于具有跳跃的过程。研究的第二个领域涉及的唯一性随机微分方程的解决方案所产生的人口模型在数学生物学。这些方程描述了随着颗粒数量的增加，每个颗粒的质量减少，以及分支率的增加，分支扩散过程的极限。允许一个粒子的分支速率和扩散机制依赖于系统中的所有其他粒子。分支扩散被用作各种物种的种群动力学模型。结果的方程通常是无限维的，退化的，或两者兼而有之。首席研究员将继续他的工作，证明这些方程的唯一性。 长期以来，人们已经知道物理和生物科学中的许多系统可以用随机过程建模。最近人们发现，许多金融和经济系统也可以这样建模。为了研究更复杂的系统，出现了新型的随机过程。举个例子，股票价格通常被视为取决于一个连续的随机过程，布朗运动。然而，股票价格的波动往往有突然的跳跃，导致战争，新的发现等，因此，它是必不可少的，也研究随机过程的跳跃。在研究种群模型时，人们期望种群的行为会根据种群的大小而有质的不同。首席研究员的研究主要涉及两种类型的随机过程，一种是跳跃的，如股票市场的例子，另一种是关于可以退化的系统的，如人口的例子。一些正在研究的问题是，是否只有一个解决方案的方程和解决方案是否有足够的规律性是有用的，在提供新的信息的模型。