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)
Ruin probabilities and penalty functions with stochastic rates of interest. (English) Zbl 1070.60043
Summary: Assume that a compound Poisson surplus process is invested in a stochastic interest process which is assumed to be a Lévy process. We derive recursive and integral equations for ruin probabilities with such an investment. Lower and upper bounds for the ultimate ruin probability are obtained from these equations. When the interest process is a Brownian motion with drift, we give a unified treatment to ruin quantities by studying the expected discounted penalty function associated with the time of ruin. An integral equation for the penalty function is given. Smooth properties of the penalty function are discussed based on the integral equation. Errors in a known result about the smooth properties of the ruin probabilities are corrected. Using a differential argument and moments of exponential functionals of Brownian motions, we derive an integro-differential equation satisfied by the penalty function. Applications of the integro-differential equation are given to the Laplace transform of the time of ruin, the deficit at ruin, the amount of claim causing ruin, etc. Some known results about ruin quantities are recovered from the generalized penalty function.

60G55Point processes
60G51Processes with independent increments; Lévy processes
60J65Brownian motion
91B30Risk theory, insurance
Full Text: DOI
[1] Asmussen, S.: Ruin probabilities. (2000) · Zbl 0960.60003
[2] Cai, J.: Ruin probabilities under dependent rates of interest. J. appl. Probab. 39, 312-323 (2002) · Zbl 1007.60096
[3] Cai, J.; Dickson, D. C. M.: On the expected discounted penalty function at ruin of a surplus process with interest. Insurance math. Econom. 30, 389-404 (2002) · Zbl 1074.91027
[4] Cai, J.; Dickson, D. C. M.: Upper bounds for ultimate ruin probabilities in the sparre andersen model with interest. Insurance math. Econom. 32, 61-71 (2003) · Zbl 1074.91028
[5] Carmona, P.; Petit, F.; Yor, M.: On exponential functionals of certain Lévy processes. Stochastics stochastic rep. 47, 71-101 (1994) · Zbl 0830.60072
[6] Dufresne, F.; Gerber, H. U.: Risk theory for the compound Poisson process that is perturbed by diffusion. Insurance math. Econom. 10, 51-59 (1991) · Zbl 0723.62065
[7] Frolova, A.; Kabanov, Y.; Pergamenshchikov, S.: In the insurance business risky investments are dangerous. Finance stoch 6, 227-235 (2002) · Zbl 1002.91037
[8] Gerber, H. U.; Shiu, E. S. W.: The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. Insurance math. Econom. 21, 129-137 (1997) · Zbl 0894.90047
[9] Grandell, J.: Aspects of risk theory. (1991) · Zbl 0717.62100
[10] Kalashnikov, V.; Norberg, R.: Power tailed ruin probabilities in the presence of risky investments. Stochastic process. Appl. 98, 211-228 (2002) · Zbl 1058.60095
[11] Norberg, R.: Ruin probabilities with assets and liabilities of diffusion type. Stochastic process. Appl. 81, 255-269 (1999) · Zbl 0962.60075
[12] Paulsen, J.: Risk theory in a stochastic environment. Stochastic process. Appl. 21, 327-361 (1993) · Zbl 0777.62098
[13] Paulsen, J.: On cramér-like asymptotics for risk processes with stochastic return on investments. Ann. appl. Probab. 12, 1247-1260 (2002) · Zbl 1019.60041
[14] Paulsen, J.; Gjessing, H. K.: Ruin theory with stochastic economic environment. Adv. appl. Probab. 29, 965-985 (1997) · Zbl 0892.90046
[15] Rolski, T.; Schmidli, V.; Schmidt, V.; Teugels, J. L.: Stochastic processes for insurance and finance. (1999) · Zbl 0940.60005
[16] Sundt, B.; Teugels, J. L.: Ruin estimates under interest force. Insurance math. Econom. 16, 7-22 (1995) · Zbl 0838.62098
[17] Wang, G.; Wu, R.: Distributions for the risk process with a stochastic return on investments. Stochastic process. Appl. 95, 329-341 (2001) · Zbl 1064.91051
[18] Willmot, G. E.; Lin, X. S.: Lundberg approximations for compound distributions with insurance applications. (2001) · Zbl 0962.62099
[19] Yor, M.: On some exponential functionals of Brownian motion. Adv. appl. Probab. 24, 509-531 (1992) · Zbl 0765.60084