Analysis of the Brownian Motion on the Matrix Lie Group SO(2) for Determining a Short-Term Interest Rate Model: A Simulation Approach
DOI:
https://doi.org/10.25077/jmua.15.2.132-149.2026Abstract
In this paper, we observe the special orthogonal matrix Lie group containing of all 2x2 real matrices, denoted by SO(2), which can be geometrically visualized as the one-dimensional torus S1 which is nothing but the unit circle. A Brownian motion on SO(2) can be constructed and represented by a stochastic differential equation defined over a dynamic state space. The research aims to derive a short-term interest rate model on SO(2) through Brownian motion analysis which is a geometric approach. We employ a qualitative methodology, including a literature review of Brownian motion, stochastic differential equations, and dynamical state-space techniques on SO(2). Firstly, we prove the isomorphism SO(2) is isomorphic to S1, secondly, we determine Brownian motion on SO(2) and its equivalent, and thirdly, we formulate the corresponding stochastic differential equation, and the last, determine the short-term interest rate equation on SO(2). In this study, it is confirmed Lim and Privault’s work that the interest rate equation on SO(2) is given by rt = beta + 2 gamma cos(Wt) with beta, gamma is constant and Wt is standard Brownian motion. To clarify the obtained results, this study also gave a quantitative approach that is Python simulation of interest rate calculation using the matrix Lie group interest rate and other equations. The interest rate equation uses the matrix Lie group SO(n) with n greater than or equal to three still open to further research that can be applied to long-term interest rates.
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