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A link between wave governed random motions and ruin processes. (English) Zbl 1103.91045
The paper links results involving wave governed random motions with problems related to the time of ruin in risk theory.
In particular, in the context of risk theory, the authors analyse problems connected to models for both positive and negative risk sums. In the first case the process $$R_t$$, representing the reserve, increases linearly with rate $$c$$, that is the strictly positive premium rate, during the time and decreases by jumps because of claims: $R_t=u+ct-S_t$ where $$R_0=u>0$$, and $$S_t$$ is the aggregate claims up to time $$t$$.
In the second case, the reserve process $$R'_t$$ decreases linearly with $$c'$$, that is the strictly positive premium rate, and increases by jumps: $R'_t=u-c't+S_t$ where $$R_0=u>0$$ and $$S'_t$$ is the aggregate jumps up to $$t$$.
In both cases, the first time where the reserve becomes negative, say $$T_u$$ and $$T'_u$$ in the case of positive and negative risk sums respectively, plays a fundamental role. In particular the finite-time ruin probabilities $$P(T_u<t)$$ and $$P(T'_u<t)$$ give relevant information in risk management. In this context the authors propose an algorithm for computing finite-time ruin probabilities in renewal non-Poisson risk models with exponential claims.
Applications of the results in finance close the paper.

##### MSC:
 91B30 Risk theory, insurance (MSC2010) 91B70 Stochastic models in economics 60G40 Stopping times; optimal stopping problems; gambling theory 35L10 Second-order hyperbolic equations 35R60 PDEs with randomness, stochastic partial differential equations
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##### References:
 [1] Di Crescenzo, A., On random motions with velocities alternating at Erlang-distributed random times, Adv. appl. probab., 33, 690-701, (2001) · Zbl 0990.60094 [2] Di Crescenzo, A.; Pellerey, F., On prices’s evolution based on alternating random processes, Appl. stoch. models business ind., 18, 171-184, (2002) [3] Foong, S.K.; Kanno, S., Properties of the telegrapher’s random process with or without a trap, Stoch. proc. appl., 53, 147-173, (1994) · Zbl 0807.60070 [4] Grandell, J., 1991. Aspects of Risk Theory, Springer Series in Statistics. · Zbl 0717.62100 [5] Kac, M., A stochastic model related to the telegrapher’s equation, Rocky mountain J. math, 4, 497-509, (1974) · Zbl 0314.60052 [6] Malinovskii, V., Non-Poissonian claim’s arrival and calculation of the probability of ruin, Insurance, math. econ., 22, 123-138, (1998) · Zbl 0907.90099 [7] Morse, P.M.; Feshbach, H., Methods of theoretical physics, (1953), MacGraw-Hill New York · Zbl 0051.40603 [8] Orsingher, E., Probability law, flow function, maximum distribution of wave-governed random motions and their connections with kirchoff’s laws, Stoch. proc. appl., 34, 49-66, (1990) · Zbl 0693.60070 [9] Panjer, H., Recursive evaluation of a family of compound distributions, Astin bull., 12, 22-26, (1981) [10] Rolski, T.; Schmidli, H.P.; Schmidt, O.; Teugels, J., Stochastic processes for insurance and finance, (2001), Wiley [11] Rullière, D., Loisel, S., 2004. Another look at the Picard-Lefèvre formula for finite-time ruin probabilities. Insurance, Math. Econ. 35, 187-203 (this issue). [12] Seal, H., Numerical calculation of the probability of ruin in the Poisson/exponential case, Mitt. verein. schweiz. versich. math., 72, 77-100, (1972) · Zbl 0274.62075 [13] Takács, L., A generalization of the ballot problem and its application in the theory of queues, J. am. stat. assoc., 57, 327-337, (1962) · Zbl 0109.36702 [14] Takács, L., Combinatorial and analytic methods in the theory of queues, Adv. appl. prob., 7, 607-635, (1975) · Zbl 0314.60069
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