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arXiv · 2501.06895

Discrete-time weak approximation of a Black-Scholes model with drift and volatility Markov switching

Vitaliy Golomoziy·Kamil Kladivko·Yuliya Mishura
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Abstract

We consider a continuous-time financial market with an asset whose price is modeled by a linear stochastic differential equation with drift and volatility switching driven by a uniformly ergodic jump Markov process with a countable state space (in fact, this is a Black-Scholes model with Markov switching). We construct a multiplicative scheme of series of discrete-time markets with discrete-time Markov switching. First, we establish that the discrete-time switching Markov chains weakly converge to the limit continuous-time Markov process. Second, having this in hand, we apply conditioning on Markov chains and prove that the discrete-time market models themselves weakly converge to the Black-Scholes model with Markov switching. The convergence is proved under very general assumptions both on the discrete-time net profits and on a generator of a continuous-time Markov switching process.

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Vitaliy Golomoziy, Kamil Kladivko, Yuliya Mishura. 2025-01-12. Discrete-time weak approximation of a Black-Scholes model with drift and volatility Markov switching. https://arxiv.org/abs/2501.06895

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