Bi-Directional Grid Constrained Stochastic Processes' Link to Multi-Skew Brownian Motion

TL;DR

This paper links Bi-Directional Grid Constrained stochastic processes to multi-skew Brownian motion, providing theoretical and simulation frameworks to analyze their properties and potential applications in finance and other fields.

Contribution

It establishes that BGCSPs are a variant of multi-skew Brownian motion, introduces a theoretical and simulation framework, and explores their complex, hidden barrier properties.

Findings

BGCSPs are a form of multi-skew Brownian motion.

Simulation results reveal complex barrier behaviors.

Applications demonstrated in finance and constrained stochastic modeling.

Abstract

Bi-Directional Grid Constrained (BGC) stochastic processes (BGCSPs) constrain the random movement toward the origin steadily more and more, the further they deviate from the origin, rather than all at once imposing reflective barriers, as does the well-established theory of It^o diffusions with such reflective barriers. We identify that BGCSPs are a variant rather than a special case of the multi-skew Brownian motion (M-SBM). This is because they have their own complexities, such as the barriers being hidden (not known in advance) and not necessarily constant over time. We provide an M-SBM theoretical framework and also a simulation framework to elaborate deeper properties of BGCSPs. The simulation framework is then applied by generating numerous simulations of the constrained paths and the results are analysed. BGCSPs have applications in finance and indeed many other fields requiring graduated constraining, from both above and below the initial position.