Optimal investment game for two regulated players with regime switching

Lin Xu; Linlin Wang; Hao Wang; Liming Zhang; Lin Xu; Linlin Wang; Hao Wang; Liming Zhang

doi:10.3934/math.20241651

AIMS Mathematics

2024, Volume 9, Issue 12: 34674-34704. doi: 10.3934/math.20241651

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Research article

Optimal investment game for two regulated players with regime switching

1.
School of Mathematics and Statistics, Anhui Normal University, Wuhu, Anhui 241002, China
2.
School of Big Data and Statistics, Anhui University, Hefei, Anhui 230601, China

Received: 14 October 2024 Revised: 04 December 2024 Accepted: 05 December 2024 Published: 12 December 2024
MSC : 91-10, 90-10, 90C39

This paper investigated a zero-sum stochastic investment game for two investors in a regime-switching market with common random time solvency regulations. We considered two types of intensities for the inter-arrival time of regulations: one was modeled as a function of a time-homogeneous Markov chain, while the other was treated as a deterministic function of time $ t $. In the first case, the associated Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation was an elliptic partial differential equation (PDE). By solving an auxiliary problem, we demonstrated the existence and regularity of the value function. In the regime-switching model, players' optimal strategies resembled those in a non-regime-switching model but required dynamic adjustments based on the Markov chain state. In the second case, the associated HJBI equation was a parabolic PDE. We provided a numerical method using a Markov chain approximation scheme and presented several numerical examples to illustrate the impact of regime switching and random time solvency on optimal policies.
- stochastic investment game,
- solvency regulations,
- fixed point method,
- Markov chain approximation
Citation: Lin Xu, Linlin Wang, Hao Wang, Liming Zhang. Optimal investment game for two regulated players with regime switching[J]. AIMS Mathematics, 2024, 9(12): 34674-34704. doi: 10.3934/math.20241651

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Abstract

This paper investigated a zero-sum stochastic investment game for two investors in a regime-switching market with common random time solvency regulations. We considered two types of intensities for the inter-arrival time of regulations: one was modeled as a function of a time-homogeneous Markov chain, while the other was treated as a deterministic function of time $ t $. In the first case, the associated Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation was an elliptic partial differential equation (PDE). By solving an auxiliary problem, we demonstrated the existence and regularity of the value function. In the regime-switching model, players' optimal strategies resembled those in a non-regime-switching model but required dynamic adjustments based on the Markov chain state. In the second case, the associated HJBI equation was a parabolic PDE. We provided a numerical method using a Markov chain approximation scheme and presented several numerical examples to illustrate the impact of regime switching and random time solvency on optimal policies.

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