zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Analysis of the expected discounted penalty function for a general jump-diffusion risk model and applications in finance. (English) Zbl 1231.91162
Summary: We extend the Cramér-Lundberg risk model perturbed by diffusion to incorporate the jumps of surplus investment return. Under the assumption that the jump of surplus investment return follows a compound Poisson process with Laplace distributed jump sizes, we obtain the explicit closed-form expression of the resulting Gerber-Shiu expected discounted penalty (EDP) function through the Wiener-Hopf factorization technique instead of the integro-differential equation approach. Especially, when the claim distribution is of phase-type, the expression of the EDP function is simplified even further as a compact matrix-type form. Finally, the financial applications include pricing barrier option and perpetual American put option and determining the optimal capital structure of a firm with endogenous default.
MSC:
91B30Risk theory, insurance
91G20Derivative securities
60J75Jump processes
References:
[1]Asmussen, S.; Avram, F.; Pistorius, M. R.: Russian and American put options under exponential phase-type Lévy models, Stochastic processes and their applications 109, 79-111 (2004) · Zbl 1075.60037 · doi:10.1016/j.spa.2003.07.005
[2]Bertoin, J.: Lévy processes, (1996) · Zbl 0861.60003
[3]Bingham, N. H.: Fluctuation theory in continuous time, Advances in applied probability 7, No. 4, 705-766 (1975) · Zbl 0322.60068 · doi:10.2307/1426397
[4]Cai, J.; Yang, H. L.: Ruin in the perturbed compound Poisson risk process under interest force, Advances in applied probability 37, 819-835 (2005) · Zbl 1074.60090 · doi:10.1239/aap/1127483749
[5]Chen, Y.T., Lee, C.F., Sheu, Y.C., 2006. On the expected discounted penalty at ruin in two-sided jump-diffusion model. National Chiao Tung University
[6]Chen, Y. T.; Lee, C. F.; Sheu, Y. C.: An ODE approach for the expected discounted penalty at ruin in a jump-diffusion model, Finance and stochastics 11, 323-355 (2007) · Zbl 1164.60034 · doi:10.1007/s00780-007-0045-5
[7]Chi, Y.; Jaimungal, S.; Lin, X. S.: An insurance risk model with stochastic volatility, Insurance: mathematics and economics (2009)
[8]Chi, Y., Lin, X.S., 2009. On threshold dividend strategy in a two-sided jump-diffusion risk model. Working paper
[9]Dufresne, F.; Gerber, H. U.: Risk theory for the compound Poisson process that is perturbed by diffusion, Insurance: mathematics and economics 10, 51-59 (1991) · Zbl 0723.62065 · doi:10.1016/0167-6687(91)90023-Q
[10]Durrett, R.: Probability: theory and examples, (1996)
[11]Gerber, H. U.; Landry, B.: On the discounted penalty at ruin in a jump-diffusion and the perpetual put option, Insurance: mathematics and economics 22, 263-276 (1998) · Zbl 0924.60075 · doi:10.1016/S0167-6687(98)00014-6
[12]Gerber, H. U.; Shiu, E. S. W.: On the time value of ruin, North American actuarial journal 2, No. 1, 48-78 (1998) · Zbl 1081.60550
[13]Hilberink, B.; Rogers, L. C. G.: Optimal capital structure and endogenous default, Finance and stochastics 6, 237-263 (2002) · Zbl 1002.91019 · doi:10.1007/s007800100058
[14]Klugman, S. A.; Panjer, H. H.; Willmot, G. E.: Loss models–from data to decisions, (2004)
[15]Kou, S. G.: A jump-diffusion model for option pricing, Management science 48, No. 8, 1086-1101 (2002) · Zbl 1216.91039 · doi:10.1287/mnsc.48.8.1086.166
[16]Kou, S. G.; Wang, H.: First passage times of a jump diffusion process, Advances in applied probability 35, 504-531 (2003) · Zbl 1037.60073 · doi:10.1239/aap/1051201658
[17]Kou, S. G.; Wang, H.: Option pricing under a double exponential jump diffusion model, Management science 50, No. 9, 1178-1192 (2004)
[18]Kyprianou, A. E.: Introductory lectures on fluctuations of Lévy processes with applications, (2006)
[19]Kyprianou, A. E.; Surya, B. A.: Principles of smooth and continuous fit in the determination of endogenous bankruptcy levels, Finance and stochastics 11, 131-152 (2007) · Zbl 1143.91020 · doi:10.1007/s00780-006-0028-y
[20]Leland, H. E.: Corporate debt value, Bond covenants, and optimal capital structure, Journal of finance 49, 1213-1252 (1994)
[21]Lin, X. S.; Willmot, G. E.: Analysis of a defective renewal equation arising in ruin theory, Insurance: mathematics and economics 25, 63-84 (1999) · Zbl 1028.91556 · doi:10.1016/S0167-6687(99)00026-8
[22]Lin, X. S.; Willmot, G. E.: The moments of the time of ruin, the surplus before ruin, and the deficit at ruin, Insurance: mathematics and economics 27, 19-44 (2000) · Zbl 0971.91031 · doi:10.1016/S0167-6687(00)00038-X
[23]Mordecki, E.: Optimal stopping and perpetual options for Lévy processes, Finance and stochastics 6, 473-493 (2002) · Zbl 1035.60038 · doi:10.1007/s007800200070
[24]Mordecki, E.: Ruin probabilities for Lévy processes with mixed-exponential negative jumps, Theory of probability and its applications 48, 170-176 (2004) · Zbl 1055.60040 · doi:10.1137/S0040585X980178
[25]Neuts, M. F.: Matrix-geometric solutions in stochastic models–an algorithmic approach, (1981)
[26]Ren, J. D.: The expected value of the time of ruin and the moments of the discounted deficit at ruin in the perturbed classical risk process, Insurance: mathematics and economics 37, 505-521 (2005) · Zbl 1129.91027 · doi:10.1016/j.insmatheco.2005.05.002
[27]Rogers, L. C. G.; Williams, D.: Diffusions, Markov processes and martingales: volume 1: foundations, (2000)
[28]Rolski, T.; Schmidli, H.; Schmidt, V.; Teugels, J.: Stochastic processes for insurance and finance, (1999) · Zbl 0940.60005
[29]Sato, K. I.: Lévy processes and infinitely divisible distributions, (1999)
[30]Tijms, H. C.: Stochastic models: an algorithmic approach, (1994) · Zbl 0838.60075