Hostname: page-component-6766d58669-zlvph Total loading time: 0 Render date: 2026-05-15T11:29:53.342Z Has data issue: false hasContentIssue false
  • English
  • Français

A stochastic differential game for quadratic-linear diffusion processes

Part of: Stochastic systems and control Stochastic processes

Published online by Cambridge University Press:  11 January 2017

Shangzhen Luo
Shangzhen Luo*
Affiliation:
University of Northern Iowa
*
* Postal address: Department of Mathematics, University of Northern Iowa, Cedar Falls, IA 50614-0506, USA. Email address: luos@uni.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Keywords

Stochastic differential gameNash equilibriumFleming–Bellman–Isaacs equationsquadratic-linear diffusion process

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 93E20: Optimal stochastic control

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 4 , December 2016 , pp. 1161 - 1182
Copyright
Copyright © Applied Probability Trust 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1] Browne, S. (2000). Stochastic differential portfolio games. J. Appl. Prob. 37, 126–147.CrossRefGoogle Scholar
[2] Elliott, R. (1976). The existence of value in stochastic differential games. SIAM J. Control Optimization 14, 85–94.CrossRefGoogle Scholar
[3] Elliott, R. and Davis, M. H. A. (1981). Optimal play in a stochastic differential game. SIAM J. Control Optimization 19, 543–554.CrossRefGoogle Scholar
[4] Emanuel, D. C., Harrison, J. M. and Taylor, A. J. (1975). A diffusion approximation for the ruin function of a risk process with compounding assets. Scand. Actuarial J. 1975, 240–247.CrossRefGoogle Scholar
[5] Fleming, W. H. and Souganidis, P. E. (1989). On the existence of value functions of two-player, zero-sum stochastic differential games. Indiana Univ. Math. J. 38, 293–314.CrossRefGoogle Scholar
[6] Guo, X. (2002). Some risk management problems for firms with internal competition and debt. J. Appl. Prob. 39, 55–69.CrossRefGoogle Scholar
[7] Guo, X., Liu, J. and Zhou, X. Y. (2004). A constrained non-linear regular-singular stochastic control problem with applications. Stoch. Process. Appl. 109, 167–187.CrossRefGoogle Scholar
[8] Krylov, N. V. (1980). Controlled Diffusion Processes, Springer, New York.CrossRefGoogle Scholar
[9] Luo, S. (2014). Stochastic Brownian game of absolute dominance. J. Appl. Prob. 51, 436–452.CrossRefGoogle Scholar
[10] Mataramvura, S. and Øksendal, B. (2008). Risk minimizing portfolios and HJBI equations for stochastic differential games. Stochastics 80, 317–337.CrossRefGoogle Scholar
[11] Meng, H., Siu, T. K. and Yang, H. (2013). Optimal dividends with debts and nonlinear insurance risk processes. Insurance Math. Econom. 53, 110–121.CrossRefGoogle Scholar
[12] Taksar, M. I. and Markussen, C. (2003). Optimal dynamic reinsurance policies for large insurance portfolios. Finance Stoch. 7, 97–121.CrossRefGoogle Scholar
[13] Taksar, M. and Zeng, X. (2011). Optimal non-proportional reinsurance control and stochastic differential games. Insurance Math. Econom. 48, 64–71.CrossRefGoogle Scholar
[14] Yeung, D. W. K. and Petrosyan, L. A. (2004). Subgame consistent cooperative solutions in stochastic differential games. J. Optimization Theory Appl. 120, 651–666.CrossRefGoogle Scholar
[15] Yeung, D. W. K. and Petrosyan, L. A. (2006). Cooperative Stochastic Differential Games. Springer, New York.Google Scholar
[16] Zeng, X. (2010). A stochastic differential reinsurance game. J. Appl. Prob. 47, 335–349.CrossRefGoogle Scholar