Stochastic Differential Equations and Related Topics

主要研究人员将研究涉及随机微分方程（以下简称SDEs）的各种问题。 这些包括期望的函数和泛函的解决方案的随机微分方程，无论是扩散或更一般的马尔可夫过程跳跃的数值近似。 受随机金融理论启发的模型将受到关注;一个例子是向前向后的SDES，其中寻求一般弱存在唯一性定理。 此外，特别强调的是专门用于过滤理论，其中一个新的蒙特卡洛方法的开发，也将试图处理无穷维的情况下，鞅问题的方法。 随机偏微分方程将通过一种新的粘性解的方法进行研究，希望在完全非线性的情况下得到解的存在性、唯一性和稳定性结果。 准线性后向随机微分方程将着眼于随机金融理论中的应用进行研究;此外，金融理论中的几个问题将直接处理，包括跳跃的完整市场和不完整市场中的竞争价格均衡。本项目主要研究随机微分方程。 微分方程对随时间变化的现象进行建模，处理变化率。 随机微分方程包括变化来自随机力的可能性。 应用的例子可以在建模无线电和X射线传输和蜂窝电话信号中找到，其中随机性来自静态噪声;病毒和植物的生长具有随机成分的生物学例子;以及在银行和金融中，利率模型，商品价格和证券价格都是用随机成分建模的。 随机微分方程是复杂和复杂的，不能用显式解来求解：所需的量需要通过使用高速计算机来近似。 需要有效的算法，而这些方法还没有被很好地理解;因此，从业者可以选择使用过于简单但方法有效的模型，或者更好地近似真实自然状态但方法效果不佳的模型。 我们建议在很大程度上通过开发适用于更好模型的算法来补救这种情况，此外，我们还建议量化模型的有效程度。

The principal investigators will study various issues involving stochastic differential equations (hereafter SDEs). These include numerical approximations of expectations of both functions and functionals of solutions of SDEs, whether they be diffusions or more generally Markov processes with jumps. Models inspired by Stochastic Finance theory will receive attention; one example is forward-backward SDEs, where a general weak existence and uniqueness theorem is sought. Further a special emphasis is devoted to filtering theory where a new Monte Carlo approach is developed, and also a martingale problem approach will be tried to treat the infinite dimensional case. Stochastic partial differential equations will be studied via a new method of viscosity solutions, hopefully yielding existence, uniqueness and stability results in the fully nonlinear case. Quasi-linear backward SDEs will be studied with an eye to applications in Stochastic Finance theory; in addition several problems in Finance theory will be treated directly, including complete markets with jumps and competitive price equilibria in incomplete markets. The principal investigators of this project study stochastic differential equations. A differential equation models phenomena that vary with time, dealing with rates of change. A stochastic differential equation includes the possibility that the change comes from random forces. Examples of applications can be found in modeling radio and x-ray transmissions, and cellular telephone signals, where the randomness comes from static noise; biological examples where growth of viruses and plants have a random component; and also in banking and finance where interest rate models, commodity prices, and securities prices are modeled with random components. The stochastic differential equations are complicated and sophisticated and cannot be solved with explicit solutions: the desired quantities need to be approximated by using a high speed computer. Efficient algorithms are needed and these methods are not yet well understood; by consequence the practitioner has a choice to use models that are too simple but with methods that work, or models that better approximate the true state of nature but for which methods work poorly, if at all. We propose to remedy this situation to a large extent by developing algorithms that work for the better models, and in addition we propose to quantify to what extent the models are effective.