A stochastic differential game for quadratic-linear diffusion processes. (English) Zbl 1357.91021

Luo, Shangzhen

Adv. Appl. Probab. 48, No. 4, 1161-1182 (2016).

Summary: In this paper we study a stochastic differential game between two insurers whose surplus processes are modelled by quadratic-linear diffusion processes. We consider an exit probability game. One insurer controls its risk process to minimize the probability that the surplus difference reaches a low level (indicating a disadvantaged surplus position of the insurer) before reaching a high level, while the other insurer aims to maximize the probability. We solve the game by finding the value function and the Nash equilibrium strategy in explicit forms.

MSC:

91B30	Risk theory, insurance (MSC2010)
91A15	Stochastic games, stochastic differential games
91A23	Differential games (aspects of game theory)
60J60	Diffusion processes
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60G40	Stopping times; optimal stopping problems; gambling theory
93E20	Optimal stochastic control

Keywords:

quadratic-linear diffusion process; stochastic differential game; insurance; Nash equilibrium; Fleming-Bellman-Isaacs equations