Low-dimensional Cox-Ingersoll-Ross process

Abstract

The present paper investigates Cox-Ingersoll-Ross (CIR) processes of dimension less than 1, with a focus on obtaining an equation of a new type including local times for the square root of the CIR process. We utilize the fact that non-negative diffusion processes can be obtained by the transformation of time and scale of some reflected Brownian motion to derive this equation, which contains a term characterized by the local time of the corresponding reflected Brownian motion. Additionally, we establish a new connection between low-dimensional CIR processes and reflected Ornstein-Uhlenbeck (ROU) processes, providing a new representation of Skorokhod reflection functions.

Parameter Estimation in Rough Bessel Model

“…For more details on the various properties of the Bessel processes, we refer the reader to Chapter XI of [11], the book [12] which considers general equations of the type (3) or [13][14][15] which deal with the case 0 < k < 1. However, in many applications, standard Brownian motion may not adequately capture the desired level of complexity observed in real-life phenomena.…”

“…Assume that τ ∈ [0, T] is a point of discontinuity of L H (and hence X H ) for some ω ∈ Ω and observe that X H (τ) = 0 since, otherwise, τ cannot be a point of discontinuity of X H and L H by (15). Let α > 0 be such that…”