zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Optimal investment for insurer with jump-diffusion risk process. (English) Zbl 1129.91020
Summary: We study optimal investment policies of an insurer with jump-diffusion risk process. Under the assumptions that the risk process is compound Poisson process perturbed by a standard Brownian motion and the insurer can invest in the money market and in a risky asset, we obtain the close form expression of the optimal policy when the utility function is exponential. We also study the insurer’s optimal policy for general objective function, a verification theorem is proved by using martingale optimality principle and Ito’s formula for jump-diffusion process. In the case of minimizing ruin probability, numerical methods and numerical results are presented for various claim-size distributions.

91B28Finance etc. (MSC2000)
49L20Dynamic programming method (infinite-dimensional problems)
60J65Brownian motion
93E20Optimal stochastic control (systems)
Full Text: DOI
[1] Asmussen, S.; Taksar, M.: Controlled diffusion models for optimal dividend pay-out. Insurance: mathematics and economics 20, 1-15 (1997) · Zbl 1065.91529
[2] Browne, Sid.: Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin. Mathematics of operations research 20, 937-958 (1995) · Zbl 0846.90012
[3] Browne, Sid.: Survival and growth with a liability: optimal portfolio strategies in continuous time. Mathematics of operations research 22, 468-493 (1997) · Zbl 0883.90011
[4] Browne, Sid.: Beating a moving target: optimal portfolio strategies for outperforming a stochastic benchmark. Finance and stochastics 3, 275-294 (1999) · Zbl 1047.91025
[5] Bühlmann, H.: Mathematical methods in risk theory. (1970) · Zbl 0209.23302
[6] Cai, J., Yang, H., 2005. Ruin in the perturbed compound Poisson risk process under interest force. Advances in Applied Probability, 37, in press. · Zbl 1074.60090
[7] Cont, R.; Tankov, P.: Financial modelling with jump processes. (2003) · Zbl 1052.91043
[8] Crandall, M. G.; Lions, P. L.: Viscosity solutions of Hamilton -- Jacobi equations. Transactions of the American mathematical society 277, 1-42 (1983) · Zbl 0599.35024
[9] Crandall, M. G.; Ishii, H.; Lions, P. L.: User’s guide to viscosity solutions of second order partial differential equations. Bulletin (new series) of the American mathematical society 27, 1-67 (1992) · Zbl 0755.35015
[10] Dayananda, P. W. A.: Optimal reinsurance. Journal of applied probability 7, 134-156 (1970) · Zbl 0191.50701
[11] Embrechts, P.; Klüppelberg, C.; Mikosch, T.: Modelling extremal events for insurance and finance. (1997)
[12] Fleming, W. H.; Rishel, R. W.: Deterministic and stochastic optimal control. (1975) · Zbl 0323.49001
[13] Fleming, W. H.; Soner, H. M.: Controlled Markov processes and viscosity solutions. (1993) · Zbl 0773.60070
[14] Gaier, J.; Grandits, P.: Ruin probabilities in the presence of regularly varying tails and optimal investment. Insurance: mathematics and economics 30, 211-217 (2002) · Zbl 1055.91049
[15] Gaier, J.; Grandits, P.: Ruin probabilities and investment under interest force in the presence of regularly varying tails. Scandinavian actuarial journal, 256-278 (2004) · Zbl 1091.62102
[16] Gerber, H.: Entscheidigungskriterien fürden zusammengesetzten Poisson prozess. Mitteilungen der vereinigung schweizer versicherungsmathematiker 19, 185-228 (1969)
[17] Gerber, H., 1979. An Introduction to Mathematical Risk Theory, Huebner Foundation Monograph No. 8.
[18] Gerber, H.; Shiu, E. S. W.: Optimal dividends: analysis with Brownian motion. North American actuarial journal 8, No. 1, 1-20 (2004) · Zbl 1085.62122
[19] Goovaerts, M. J.; Kass, R.; Van Heerwarrden, A. E.; Bauwelinckx, T.: Effective actuarial methods. (1990)
[20] Hipp, C.; Plum, M.: Optimal investment for insurers. Insurance: mathematics and economics 27, 215-228 (2000) · Zbl 1007.91025
[21] Hipp, C.; Plum, M.: Optimal investment for investors with state dependent income, and for insurers. Finance and stochastics 7, 299-321 (2003) · Zbl 1069.91051
[22] Hipp, C.; Taksar, M.: Stochastic control for optimal new business. Insurance: mathematics and economics 26, 185-192 (2000) · Zbl 1103.91366
[23] Huang, C. H.; Litzenberger, R. H.: Foundations for financial economics. (1987) · Zbl 0677.90001
[24] Højgaard, B.; Taksar, M.: Optimal proportional reinsurance policies for diffusion models. Scandinavian actuarial journal 2, 166-180 (1998) · Zbl 1075.91559
[25] Højgaard, B.; Taksar, M.: Optimal proportional reinsurance policies for diffusion models with transaction costs. Insurance: mathematics and economics 22, 41-51 (1998) · Zbl 1093.91518
[26] Højgaard, B., Taksar, M., 2000. Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy. Preprint. · Zbl 0958.91026
[27] Irgend, C.; Paulsen, J.: Optimal control of risk exposure, reinsurance and investments for insurance portfolios. Insurance: mathematics and economics 35, 21-51 (2004) · Zbl 1052.62107
[28] Jeanblanc-Picqué, M.; Shiryaev, A. N.: Optimization of the flow of dividends. Russian mathematical surveys 50, 257-277 (1995) · Zbl 0878.90014
[29] Karatzas, I.; Shreve, S.: Brownian motion and stochastic calculus. (1988) · Zbl 0638.60065
[30] Krylov, N. V.: Controlled diffusion processes. (1980) · Zbl 0459.93002
[31] Liu, C. S.; Yang, H.: Optimal investment for a insurer to minimize its probability of ruin. North American actuarial journal 8, No. 2, 11-31 (2004) · Zbl 1085.60511
[32] Martin-Löf, A.: A method for finding the optimal decision rule for a policy holder of an insurance with a bonus system. Scandinavian actuarial journal, 23-39 (1973) · Zbl 0322.62102
[33] Martin-Löf, A.: Premium control in an insurance system, an approach using linear control theory. Scandinavian actuarial journal, 1-27 (1983) · Zbl 0509.62097
[34] Martin-Löf, A.: Lectures on the use of control theory in insurance. Scandinavian actuarial journal, 1-25 (1994) · Zbl 0802.62090
[35] Moore, K.S., Young, V.R., 2004. Optimal insurance in a continuous-time model. Preprint.
[36] Milevsky, M.A., Moore, K.S., Young, V.R., 2004. Optimal asset allocation and ruin minimization annuitization strategies. Preprint.
[37] Paulsen, J.: Optimal dividend payouts for diffusions with solvency constraints. Finance and stochastics 4, 457-474 (2003) · Zbl 1038.60081
[38] Paulsen, J.; Gjessing, H. K.: Optimal choice of dividend barriers for a risk process with stochastic return on investments. Insurance: mathematics and economics 20, 215-223 (1997) · Zbl 0894.90048
[39] Schmidli, H.: On minimizing the ruin probability by investment and reinsurance. Annals of applied probability 12, 890-907 (2002) · Zbl 1021.60061
[40] Schmidli, H.: On optimal investment and subexponential claims. Insurance: mathematics and economics 36, 25-35 (2005) · Zbl 1110.91019
[41] Taksar, M.: Optimal risk and dividend distribution control for an insurance company. Mathematical methods of operations research 1, 1-42 (2000) · Zbl 0947.91043
[42] Taksar, M.; Markussen, C.: Optimal dynamic reinsurance policies for large insurance portfolios. Finance and stochastics 7, 97-121 (2003) · Zbl 1066.91052
[43] Wang, G.; Wu, R.: Distributions for risk process with a stochastic return on investment. Stochastic processes and their application 95, 329-341 (2001) · Zbl 1064.91051