Collaborative Research: Tolerance-Enforced Simulation of Stochastic Processes

Our research goal is to investigate an innovative class of stochastic simulation procedures which strongly approximate a wide range of random processes and associated expectations of sample-path functionals. We call this class of procedures Tolerance-Enforced Simulation (TES) algorithms. These algorithms enable the approximation of a fully continuous random objects, for example, Gaussian random fields, Levy-driven SDEs by a piecewise linear process with a user-defined (deterministic) error in various path-dependent norms with 100% certainty. This proposal explains how the construction of these types of procedures can be obtained, optimized, and how such constructions can be applied. Our strategy consists in constructing optimal TES under a wide range of useful metrics (including the uniform norm) in path space. The selection of appropriate metric is motivated by a class of functionals of interests. The idea is to transfer the TES algorithm from a fundamental process, such as Brownian motion, to a more complex one. For example, TES algorithms for SDEs can be obtained from TES algorithms for Brownian motion, but in this example the metric of interest requires utilizing the theory of rough paths. Intellectual merit: Our research program provides a completely different approach to the numerical analysis of continuous random structures. To make the point, let us once again consider the case of SDEs. The standard way of approximating SDEs is by applying an Euler discretization procedure, which induces a random error. Our procedure delivers also an Euler type discretization, but the size of the grid is random and carefully constructed so that the error is deterministic and defined by the user. We plan to investigate TES algorithms for a wide range of fundamental processes. We stress that TES have important implications. For example, when using a TES procedure, the error analysis often becomes an exercise in analysis (no probability is needed). One just needs to understand the continuity properties implied by the metric underlying the TES procedure and the function whose expectation one is interested in computing. Another implication that we discuss involves exact simulation of stochastic processes. We show how TES algorithms immediately imply the construction of unbiased estimators for very general path dependent functions. We believe that typical TES procedures, which deliver a significantly stronger error control than standard methods, can be constructed at basically the same computational cost than standard methods and we plan to investigate the optimality of the procedures. Broader impact: The broader impact can be evaluated in the following dimensions: human resource development (training of PhD students), recruitment of under-represented groups, dissemination, and impact in engineering and other scientific areas. We plan to involve two PhD students, one working at Columbia and the other at Northwestern. A new PhD course, jointly designed by the PIs on the topic of this proposal, will be developed and the course material will be made available on-line. The scientific output has the potential to substantially impact areas such as applied mathematics (due to the proposal's projection on numerical methods), engineering (given the motivating applications discussed in the body of the proposal) and computational statistics in the context of inference of continuous objects. We will attempt to recruit high-quality personnel from