Complex Stochastic Systems and the Effect of Discretization

在各种物理和生物系统的行为中普遍存在随机性是公认的。例如，细胞过程，股票价格波动，天气模式，微观粒子的运动表现出不同类型的随机行为。此外，随机性是数据科学中流行的大多数现代算法的重要组成部分。 因此，它是至关重要的，各种类型的随机性的特点是正确的详细了解系统的属性。包含这种随机性的数学模型通常用随机方程表示，这些方程通常具有复杂的动态特性。 这些方程的数学研究的一个重要组成部分包括它们的时间演化的计算机模拟，这对于理解它们随时间的行为模式是必要的。从观测数据中对这些随机方程的一些关键参数进行准确的统计估计也是至关重要的，因为它会导致这些数据点背后的“最合适的数学模型”。这反过来又导致这些模型的预测能力更好。该奖项支持的研究将对用于这些目的的各种数值方案进行适当的数学分析。更具体地说，这一方向的研究将涉及精确计算从这些方案中获得准确结果的概率，并将描述这些概率非常高的条件。有这样的理论结果的关键需求，因为它们将不可避免地导致更快，更有效的算法，这反过来将有利于社会的设计。该项目将涉及本科生和研究生，并将帮助他们获得宝贵的分析和计算技能。该项目的成果将发表在著名的科学期刊上，并将在国内和国际会议上发表。 研究将集中在复杂随机系统的极限行为离散适当缩放步长。离散化是各种数值格式的核心，但其对所需收敛性的影响并不总是清楚。例如，众所周知，通过从Euler-Maruyama型离散化方案（使用固定步长）获得的样本路径的时间平均值来近似遍历随机微分方程（ELS）的平稳分布可能是有问题的。特别是，这样的估计将有偏差，这可能是或可能不是量化的。这些都是无限时间的地平线问题，并有一个赤字适当的渐近结果在这个方向上，即使是经常伊藤扩散。探索适当的缩放技术，以获得所需的收敛沿着与收敛速度，这样的离散为基础的计划是这个项目的中心目标。随机模型，将被视为涵盖切换跳跃扩散模型，多尺度系统和系统的相互作用的可能跳跃的SDES。最后一类是特别重要的理解粒子为基础的方法近似非线性积分微分方程，包括玻尔兹曼型方程。这些方程连接到适当的系统的相互作用的SDES跳跃通过McKean-Vlasov型限制。感兴趣的随机方程还包括Langevin随机微分方程，它对存在粒子相互作用势、阻尼和随机力的分子系统的动力学进行建模，并且在某些马尔可夫链蒙特卡罗算法中也发挥着关键作用。 该研究特别强调误差概率的大偏差渐近性，重要的是，由于存在上界和下界，给出了最佳指数衰减率。 该项目由概率计划和刺激竞争研究的既定计划（EPSCoR）共同资助。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Ubiquitous presence of randomness in behaviors of various physical and biological systems is universally acknowledged. For example, cellular processes, fluctuations in stock prices, weather patterns, movement of microscopic particles exhibit different types of random behaviors. Furthermore, randomness is a vital ingredient in most modern algorithms that are popular in data-science. Hence, it is of fundamental importance that various types of randomness are properly characterized for detailed understanding of system properties. Mathematical models incorporating such randomness are typically expressed in terms of stochastic equations, which often have complex dynamics. An important component of mathematical studies of these equations includes computer simulations of their temporal evolution, which are necessary for understanding their behavioral patterns over time. Accurate statistical estimation of some key parameters of these stochastic equations from observed data is also crucial, as it leads to the "most appropriate mathematical model" underlying these data points. This in turn leads to better predictive power of such models. The research supported by this award will undertake proper mathematical analyses of various numerical schemes that are used for these purposes. More specifically, research in this direction will involve precise calculations of probabilities of getting accurate results from these schemes and will describe conditions under which these probabilities are very high. There is a critical need for such theoretical results, since they will inevitably lead to design of faster and more efficient algorithms, which in turn will be beneficial to society. The project will involve undergraduate and graduate students and will help them to gain valuable analytical and computational skills. The results of the project will be published in well-known scientific journals and will also be presented at domestic and international conferences. The research will focus on the limiting behaviors of complex stochastic systems discretized by properly scaled step sizes. Discretization is at the heart of various numerical schemes, but its effect on the desired convergence properties is not always clearly understood. For example, it is well known that approximating stationary distribution of an ergodic stochastic differential equation (SDE) by time averages of sample paths obtained from an Euler-Maruyama type discretization scheme (using fixed step-size) could be problematic. In particular, such an estimator will have a bias, which may or may not be quantifiable. These are infinite-time horizon problems, and there is a deficit of proper asymptotic results in this direction even for regular Ito-diffusions. Exploring proper scaling techniques to get desired convergence along with convergence rates for such discretization-based schemes is the central goal of this project. Stochastic models that will be considered covers switching jump-diffusion models, multiscale systems and systems of interacting SDEs with possible jumps. The last class is particularly important for understanding particle-based methods for approximating nonlinear integro-differential equations including Boltzmann-type equations. These equations are connected to appropriate systems of interacting SDEs with jumps through McKean-Vlasov type limits. The stochastic equations of interest will also include Langevin SDEs, which model dynamics of molecular systems in presence of particle interaction potential, damping and random forces, and which also play a pivotal role in certain Markov Chain Monte Carlo algorithms. The research puts special emphasis on large deviation asymptotics of error probabilities, which, importantly, because of presence of both upper and lower bounds, give the optimal exponential decay rate. This project is jointly funded by Probability program and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.