Optimal reinsurance via Dirac-Feynman approach. (English) Zbl 1429.91284

Summary: In this paper, the Dirac-Feynman path calculation approach is applied to analyse finite time ruin probability of a surplus process exposed to reinsurance by capital injections. Several reinsurance optimization problems on optimum insurance and reinsurance premium with respect to retention level are investigated and numerically illustrated. The retention level is chosen to decrease the finite time ruin probability and to guarantee that reinsurance premium covers an average of overall capital injections. All computations are based on Dirac-Feynman path calculation approach applied to the convolution type operators perturbed by Injection operator (shift type operator). In addition, the effect of the Injection operator on ruin probability is analysed.


91G05 Actuarial mathematics
62P05 Applications of statistics to actuarial sciences and financial mathematics
58D30 Applications of manifolds of mappings to the sciences
Full Text: DOI


[1] Asmussen S, Albrecher H (2010) Ruin probabilities. World Scientific, Singapore · Zbl 1247.91080
[2] Baaquie BE (2007) Quantum finance: Path integrals and Hamiltonians for options and interest rates. Cambridge University press, Cambridge · Zbl 1140.91035
[3] Castaner A, Claramunt MM, Lefevre C (2013) Survival probabilities in bivariate risk models, with application to reinsurance. Insur: Math Econ 53:632-642 · Zbl 1290.91077
[4] Denuit M, Vermandele C (1998) Optimal reinsurance and stop-loss order. Insur: Math Econ 22:229-233 · Zbl 0986.62085
[5] Dickson DC, Waters HR (1996) Reinsurance and ruin. Insur: Math Econ 19:61-80 · Zbl 0894.62110
[6] Dickson DC, Waters HR (2004) Some optimal dividends problems. Astin Bullet 34:49-74 · Zbl 1097.91040
[7] Eisenberg J, Schmidli H (2011) Minimising expected discounted capital injections by reinsurance in a classical risk model. Scand Actuar J 2011(3):155-176 · Zbl 1277.60145
[8] Kamae T, Krengel U, O’Brien GL (1977) Stochastic inequalities on partially ordered spaces. Ann Probab 5(6):899-912 · Zbl 0371.60013
[9] Lefevre C, Loisel S (2008) On finite-time ruin probabilities for classical risk models. Scand Actuar J 2008(1):41-60 · Zbl 1164.91033
[10] Picard P, Lefevre C (1997) The probability of ruin in finite time with discrete claim size distribution. Scand Actuar J 1997(1):58-69 · Zbl 0926.62103
[11] Nie C, Dickson DC, Li S (2011) Minimizing the ruin probability through capital injections. Ann Actuar Sci 5:195-209
[12] Nie C, Dickson DC, Li S (2015) The finite time ruin probability in a risk model with capital injections. Scandinavian Actuarial Journal 2015(4):301-318 · Zbl 1398.91350
[13] Rulliere D, Loisel S (2004) Another look at the Picard-Lefevre formula for finite-time ruin probabilities. Insur: Math Econ 35:187-203 · Zbl 1103.91048
[14] Schmidli H (2002) On minimizing the ruin probability by investment and reinsurance. Ann Appl Probab 12:890-907 · Zbl 1021.60061
[15] Tamturk M, Utev S (2018) Ruin probability via Quantum Mechanics Approach. Insur: Math Econ 79:69-74 · Zbl 1401.91197
[16] Zhou M, Yuen KC (2012) Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle. Econ Model 29:198-207
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.