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SDEs的函数和解的泛函的期望的数值近似，无论它们是扩散过程还是更一般的具有跳跃的马尔可夫过程。受随机金融理论启发的模型将受到关注；一个例子是正-倒向SDEs，其中寻求一般弱存在唯一性定理。此外，特别强调了滤波理论，其中发展了一种新的蒙特卡罗方法，并尝试了一种鞅问题方法来处理无限维情况。本文将通过一种新的粘性解方法来研究随机偏微分方程，以期在完全非线性情况下得到存在性、唯一性和稳定性的结果。拟线性倒向SDEs将着眼于在随机金融理论中的应用；此外，还将直接讨论金融理论中的几个问题，包括具有跳跃的完全市场和不完全市场中的竞争性价格均衡。这个项目的主要研究者研究随机微分方程。微分方程为随时间变化的现象建模，处理变化率。随机微分方程包含了变化来自随机力的可能性。应用的例子可以在无线电和x射线传输和蜂窝电话信号的建模中找到，其中的随机性来自静态噪声；在生物学的例子中，病毒和植物的生长具有随机成分；在银行和金融领域，利率模型，商品价格和证券价格都是随机组成部分。随机微分方程是复杂而精密的，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，而这些方法尚未得到很好的理解；因此，从业者可以选择使用过于简单但方法有效的模型，或者使用更接近自然真实状态的模型，但方法效果很差（如果有的话）。我们建议通过开发适用于更好模型的算法来在很大程度上纠正这种情况，此外，我们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.