Iterated Logarithm Bounds of BGC Stochastic Processes

TL;DR

This paper introduces a new class of stochastic processes called Bi-Directional Grid Constrained (BGC) processes, providing novel bounds based on a modified Law of Iterated Logarithm, with simulations demonstrating their properties and potential applications.

Contribution

It presents a new BGC framework and a modified LIL theorem to analyze bounds of these processes, with simulations illustrating their behavior.

Findings

BGC processes are constrained based on their distance from the origin.

Modified LIL provides bounds for BGC processes.

Simulations reveal properties useful for financial modeling.

Abstract

We derive a novel framework called Bi-Directional Grid Constrained (BGC) stochastic processes in which the further an Ito diffusion drifts away from the origin, then the further it will be constrained. By making suitable modifications to the Law of Iterated Logarithm (LIL), we derive a novel theorem about the upper and lower bounds for BGC processes and their hidden barrier. To visualize the theorem, we run many simulations of the Ito diffusions for a representative expression for lambda(X, t), both before and after BGC and uncover some interesting results with applications into finance and many other areas.