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A direct approach to a first-passage problem with applications in risk theory. (English) Zbl 1232.91350
The authors consider a reflected version of the classical surplus process used in ruin theory. This process constitutes a resonable attempt to model the surplus of companies with steady outflows and sporadic inflows (e. g. discoveries, patents). Pharmaceutical or petroleum companies are prime examples of such companies. For this type of process, a risk management policy is implemented to reduce the expense rate if no inflow is generated within an Erlang-$$n$$ time period. To define the surplus process of interest, the authors introduce a Markovian representation. A homogeneous linear integro-differential equation for the Laplace transform of the time to ruin is derived. Boundary conditions of this equation are used to complete the Laplace transform representation. Numerical applications are presented to show that the considered budget reduction policy can be an effective risk management tool.

##### MSC:
 91B30 Risk theory, insurance (MSC2010) 60K15 Markov renewal processes, semi-Markov processes 60K20 Applications of Markov renewal processes (reliability, queueing networks, etc.) 60K37 Processes in random environments 60J75 Jump processes (MSC2010)
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##### References:
 [1] Agnew J.L., Linear Algebra with Applications (1989) [2] Asmussen S., Astin Bulletin 32 pp 267– (2002) · Zbl 1081.60028 · doi:10.2143/AST.32.2.1029 [3] Avanzi B., Insurance: Mathematics and Economics 41 pp 111– (2007) · Zbl 1131.91026 · doi:10.1016/j.insmatheco.2006.10.002 [4] Avram F., Insurance: Mathematics and Economics 32 pp 371– (2003) · Zbl 1074.91026 · doi:10.1016/S0167-6687(03)00117-3 [5] Bühlmann H., Mathematical Methods in Risk Theory (1970) · Zbl 0209.23302 [6] Cohen J.W., The Single Server Queue (1969) · Zbl 0183.49204 [7] Cramér H., Collective Risk Theory: A Survey of the Theory from the Point of View of the Theory of Stochastic Processes (1955) [8] Grandell J., Aspects of Risk Theory (1991) · Zbl 0717.62100 [9] Ramaswami V., Methodology and Computing in Applied Probability 8 pp 497– (2006) · Zbl 1110.60067 · doi:10.1007/s11009-006-0426-9 [10] Rolski T., Stochastic Processes for Insurance and Finance (1999) · Zbl 0940.60005 · doi:10.1002/9780470317044 [11] Seal H.L., Stochastic Theory of a Risk Business (1969) · Zbl 0196.23501 [12] Stanford D.A., Astin Bulletin 35 pp 131– (2005) · Zbl 1123.62078 · doi:10.2143/AST.35.1.583169 [13] Subramanian V., Queueing Systems 34 pp 215– (2000) · Zbl 0942.90019 · doi:10.1023/A:1019161120564 [14] Takacs L., Combinatorial Methods in the Theory of Stochastic Processes (1967) · Zbl 0162.21303 [15] Takine T., Stochastic Models 10 pp 183– (1994) · Zbl 0791.60088 · doi:10.1080/15326349408807292
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