Inference for Diffusions and Related Processes

传统的扩散模拟方法和相关的蒙特卡罗方法依赖于时间离散化技术。这种方法有两个明显的缺点：它通常是近似的，时间增量通常需要很小，以确保适当的近似，因此方法可以是计算昂贵的。最近的新方法，这个问题已经规避了需要disretise时间使用一个强大的和灵活的新的模拟思想称为回溯采样。这种方法产生精确的模拟（任何用于实验的计算机的精度约束），并具有显着的效率属性，所以在这种情况下，似乎没有成本的准确性。然而，精确算法（EA）框架只能应用于某些类别的扩散过程（尽管这类基本上包括所有一维非爆炸扩散）。这种方法既可以与纯模拟方法学一起使用，也可以与相关的（更灵活的）重要性抽样技术一起使用。我们将重点关注两个重要领域。扩散相关模型在系统生物学中的应用正在迅速扩大，我们将在这里应用我们的方法。其次，我们将考虑分子动力学模拟中的稀有事件模拟问题。

Traditional methods for diffusion simulation and related Monte Carlo methods have relied on time-discretisation techniques. This approach has two significant disadvantages: it is usually approximate, and time increments typically need to be small to ensure adequacy of the approximation, and thus methods can be computationally expensive.Recent new methodology for this problem has circumvented the need to disretise time by the use of a powerful and flexible new simulation idea known as Retrospective Sampling. This methodology produces exact simulations (to the accuracy constraints of any computer used for the experiment) and has remarkable efficiency properties, so that there appears to be no cost for exactness in this case. However the Exact Algorithm (EA) framework can be applied only for certain classes of diffusion processes (although this class essentially includes all one-dimensional non-explosive diffusions)This project aims to extend the framework above to a very rich and diverse class of stochastic processes, such as jump diffusions, hypo-elliptic diffusions and solutions of stochastic partial differential equations). The approach is to work both with pure simulation methodology and also with related (and more flexible) importance sampling techniques.There are many potential applications of these methods in scientific problems. We will focus on two important areas. The use of diffusion-related models in Systems Biology is expanding rapidly, and we will apply our methodology here. Secondly, we will consider the problem of rare event simulation in molecular dynamics simulation.