# Project Details

### Description

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

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

Status | Active |
---|---|

Effective start/end date | 9/1/17 → 8/31/20 |

### Funding

- National Science Foundation (DMS-1720433)