A discrete-time model that weakly converges to a continuous-time geometric Brownian motion with Markov switching drift rate - Golomoziy, Vitaliy; Mishura, Yuliya; Kladívko, Kamil (Frontiers in Applied Mathematics and Statistics, 2024)
This paper is devoted to studying a geometric Brownian motion with drift switching driven by 2 × 2 Markov chain. A discrete-time multiplicative approximation scheme was developed and its convergence in Skorokhod topology to the continuous-time geometric Brownian motion with switching has been proved. Furthermore, in a financial market where the discounted asset price follows a geometric Brownian motion with drift switching, market incompleteness was established, and multiple equivalent martingale measures were constructed.