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On the gain of collaboration in a two dimensional ruin problem. (English) Zbl 1446.91061

The author studies the best strategy for two insurance companies which collaborate by transfer payments to ensure survival of both. By assumption their endowment processes are independent Brownian processes with positive drifts and the transfer process is absolutely continuous. Searching the solution leads to an optimal control problem and its associated Hamilton-Jacobi-Bellman equation. An explicit formula is obtained for the value function of the problem and it appears that the optimal strategy amounts to giving all possible drift to the weaker company, apart from some set reserve.

MSC:

91G05 Actuarial mathematics
60G44 Martingales with continuous parameter
93E20 Optimal stochastic control
91G80 Financial applications of other theories
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[1] Asmussen S, Albrecher H (2010) Ruin probabilities, vol 14, 2nd edn. Advanced series on statistical science and applied probability. World Scientific Publishing, Hackensack · Zbl 1247.91080 · doi:10.1142/7431
[2] Albrecher H, Azcue P, Muler N (2017) Optimal dividend strategies for two collaborating insurance companies. Adv Appl Probab 49(2):515-548 · Zbl 1429.91274 · doi:10.1017/apr.2017.11
[3] Avanzi B (2009) Strategies for dividend distribution: a review. N Am Actuar J 13(2):217-251 · Zbl 1483.91177 · doi:10.1080/10920277.2009.10597549
[4] Avram F, Palmowski Z, Pistorius M (2008) Exit problem of a two-dimensional risk process from the quadrant: exact and asymptotic results. Ann Appl Probab 18(6):2421-2449 · Zbl 1163.60010 · doi:10.1214/08-AAP529
[5] Collamore J (1996) Hitting probabilities and large deviations. Ann Probab 24(4):2065-2078 · Zbl 0879.60021 · doi:10.1214/aop/1041903218
[6] Gerber HU, Shiu E (2006) On the merger of two companies. N Am Actuar J 10(3):60-67 · Zbl 1480.91306 · doi:10.1080/10920277.2006.10597403
[7] Grandits P (2019) A two dimensional dividend problem for collaborating companies and an optimal stopping problem. Scand Actuar J 2019(1):80−96 · Zbl 1418.91239 · doi:10.1080/03461238.2018.1498387
[8] Gu J, Steffensen M, Zheng H (2018) Optimal dividend strategies of two collaborating businesses in the diffusion approximation model. Math Oper Res 43(2):377-398 · Zbl 1435.91151 · doi:10.1287/moor.2017.0860
[9] McKean HP, Shepp LA (2006) The advantage of capitalism vs. socialism depends on the criterion. J Math Sci 139(3):6589-6594 · Zbl 1121.91068 · doi:10.1007/s10958-006-0374-5
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