Convergence of Markovian price processes in a financial market transaction model

This paper studies a financial market transaction model and convergence of Markovian price processes generated by an $\mathit{\alpha }$ ‐double auction in Xu et al. (Expert Syst Appl 41(16):7032–7045, 2014) and extends their results for a fixed $\mathit{\alpha }$ in [0, 1] to the case where $\mathit{\alpha }$ is governed by a time non‐homogeneous Markov chain over a set of finite states defined by $R\equiv \{{\mathit{\alpha }}_1,{\mathit{\alpha }}_2,\dots ,{\mathit{\alpha }}_r\}$ , $0\le {\mathit{\alpha }}_1<{\mathit{\alpha }}_2<\dots <{\mathit{\alpha }}_r\le 1$ . A convergence result similar to that in Xu et al. (2014) holds, with the fixed $\mathit{\alpha }$ replaced with the average ${\mathit{\alpha }}^{\ast }=\frac{1}{r}{\sum }_{\mathit{\theta }=1}^r{\mathit{\alpha }}_{\mathit{\theta }}$ . We also identify the conditions under which a price process generated by such a Markovian $\mathit{\alpha }$ ‐double auction converges in probability to a Walrasian equilibrium of the underlying financial market transaction model. A number of simulations are conducted and these simulations are consistent with the theoretical results of the paper.