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SDE)的各种问题。这包括对随机微分方程的函数和解的泛函的期望的数值逼近，无论它们是扩散过程还是更一般的带跳的马尔可夫过程。受到随机金融理论启发的模型将受到关注；一个例子是前向后向SDE，它寻求一个普遍的弱存在唯一性定理。此外，还特别强调了过滤理论，其中发展了一种新的蒙特卡罗方法，并尝试了一种用于处理无限维情形的鞅问题方法。利用粘性解的一种新方法研究随机偏微分方程解的存在性、唯一性和稳定性。拟线性倒向随机经济系统的研究将着眼于随机金融理论的应用；此外，还将直接处理金融理论中的几个问题，包括带跳跃的完全市场和不完全市场中的竞争价格均衡。这个项目的主要研究人员研究随机微分方程式。微分方程式模拟随时间变化的现象，处理变化率。随机微分方程包含变化来自随机力的可能性。应用的例子有：模拟无线电和X射线传输，以及蜂窝电话信号，其中随机性来自静态噪声；生物学例子，其中病毒和植物的生长具有随机成分；也可以在银行和金融中找到应用，其中利率模型、商品价格和证券价格用随机成分建模。随机微分方程组是复杂复杂的，不能用显式解来求解：需要用高速计算机来逼近所需的量。需要有效的算法，而这些方法还没有被很好地理解；因此，实践者可以选择使用过于简单但方法有效的模型，或者使用更接近自然真实状态但方法效果不佳的模型。我们建议通过开发适用于更好的模型的算法来在很大程度上纠正这种情况，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.