, order to apply it, one however needs to verify the following points 1. that a n =, p.25

. =-q-tn, Another aspect in [GR07] is another optimization problem that involves how to optimally send packets of fluid data to two queues that satisfy Equation (3.2). To conclude this section, we will present how Theorem 15 translates to a risk theory framework. Let a = {a n , n ? } and {N t (a), t ? 0} be defined as before, and consider the following risk process 1. We have exhibited a martingale related to M n (u) which is different from Biggin's martingale. By standard positive martingale and bounded submartingale theory, underline that Q t (a) only depends on

, For the moment, we are trying to find how to obtain some expression of its expectation E(M ? (u)) by the standard method of conditioning with respect to the state of the branching random walk at generation n = 1, We are trying to give some characteristic on M ? (u)

Ë. , Ê. , and Ë. , Two kinds of sets Ë have been studied in the above points

, In the case Ë = Ì, distribution ov the V n 's may be assumed constant

S. Asmussen, Albrecher -Ruin probabilities, second éd, Advanced Series on Statistical Science & Applied Probability, vol.14, 2010.

S. Asmussen, F. Avram, and &. , Usabel -Erlangian approximations for finitehorizon ruin probabilities, Astin Bull, vol.32, issue.2, pp.267-281, 2002.

E. Altman and B. , Gaujal & A. Hordijk -Discrete-event control of stochastic networks : multimodularity and regularity, Lecture Notes in Mathematics, vol.1829, 2003.

S. Asmussen, Kella -Rate modulation in dams and ruin problem, Journal of Applied Probability, vol.33, issue.2, pp.523-535, 1996.

S. Asmussen, Kella -A multi-dimensional martingale for Markov additive processes and its applications, Adv. in Appl. Probab, vol.32, issue.2, pp.376-393, 2000.

F. Avram, N. Leonenko, and &. , Rabehasaina -Series expansions for the first passage distribution of Wong-Pearson jump-diffusions, Stoch. Anal. Appl, vol.27, issue.4, pp.770-796, 2009.

F. Avram, Z. Palmowski, and &. , Pistorius -A two-dimensional ruin problem on the positive quadrant, Insurance Math. Econom, vol.42, issue.1, pp.227-234, 2008.

F. Avram and Z. &. Palmowski, Pistorius -Exit problem of a twodimensional risk process from the quadrant : exact and asymptotic results, Ann. Appl. Probab, vol.18, issue.6, pp.2421-2449, 2008.

S. Asmussen, Applied probability and queues, second éd, Stochastic Modelling and Applied Probability, vol.51, 2003.

S. Asmussen, Schock Petersen -Ruin probabilities expressed in terms of storage processes, Adv. in Appl. Probab, vol.20, issue.4, pp.913-916, 1988.

H. Albrecher and J. L. Teugels-&-r, Tichy -On a gamma series expansion for the time-dependent probability of collective ruin, 4th IME Conference, vol.29, pp.345-355, 2000.

E. J. , Baurdoux -Last exit before an exponential time for spectrally negative Lévy processes, J. Appl. Probab, vol.46, issue.2, pp.542-558, 2009.

A. Badescu and &. Breuer, The use of vector-valued martingales in risk theory, Bl. DGVFM, vol.29, issue.1, pp.1-12, 2008.

]. A. +-05, L. Badescu, A. Breuer, . Da-silva, G. Soares et al., Remiche & D. Stanford -Risk processes analyzed as fluid queues, Scand. Actuar. J, issue.2, pp.127-141, 2005.

A. L. Badescu and E. C. Cheung-&-l, Rabehasaina -A two-dimensional risk model with proportional reinsurance, J. Appl. Probab, vol.48, issue.3, pp.749-765, 2011.

J. , Bertoin -Lévy processes, Cambridge Tracts in Mathematics, vol.121, 1996.

R. Biard, Asymptotic multivariate finite-time ruin probabilities with heavy-tailed claim amounts : Impact of dependence and optimal reserve allocation, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00538571

E. E. Biffis-&-a, Kyprianou -A note on scale functions and the time value of ruin for Lévy insurance risk processes, Insurance Math. Econom, vol.46, issue.1, pp.85-91, 2010.

R. Biard, S. Loisel, C. Macci, and &. , Veraverbeke -Asymptotic behavior of the finite-time expected time-integrated negative part of some risk processes and optimal reserve allocation, J. Math. Anal. Appl, vol.367, issue.2, pp.535-549, 2010.

O. J. Boxma, A. Löpker, and &. D. Perry, Threshold strategies for risk processes and their relation to queueing theory, New frontiers in applied probability : a Festschrift for Soren Asmussen, vol.48, pp.29-38, 2011.

C. Barker and &. Newby, Optimal non-periodic inspection for a multivariate degradation model, Reliab Eng Syst Saf, vol.94, issue.1, pp.33-43, 2009.

K. Burnecki, Self-similar processes as weak limits of a risk reserve process, Probab. Math. Statist, vol.20, issue.2, pp.261-272, 2000.

J. Cai, H. U. Gerber, and &. H. Yang, Optimal dividends in an Ornstein-Uhlenbeck type model with credit and debit interest, N. Am. Actuar. J, vol.10, issue.2, pp.94-119, 2006.

E. C. Cheung-&-d, Landriault -A generalized penalty function with the maximum surplus prior to ruin in a MAP risk model, Insurance Math. Econom, vol.46, issue.1, pp.127-134, 2010.

H. D. Chen-&-d and . Yao, Fundamentals of queueing networks, 2001.

S. N. Chiu-&-c, Yin -Passage times for a spectrally negative Lévy process with applications to risk theory, Bernoulli, vol.11, issue.3, pp.511-522, 2005.

B. Auria, J. Ivanovs, O. Kella, and &. , Mandjes -Two-sided reflection of Markov-modulated Brownian motion, Stoch. Models, vol.28, issue.2, pp.316-332, 2012.

K. Debicki, K. M. Kosi?ski, M. Mandjes, and &. , Rolski -Extremes of multidimensional Gaussian processes, Stochastic Process. Appl, vol.120, issue.12, pp.2289-2301, 2010.

L. Decreusefond, Nualart -Hitting times for Gaussian processes, Ann. Probab, vol.36, issue.1, pp.319-330, 2008.

N. G. Duffield-&-n.-o', Connell -Large deviations and overflow probabilities for the general single-server queue, with applications, Math. Proc. Cambridge Philos. Soc, vol.118, issue.2, pp.363-374, 1995.

D. C. Dickson-&-g, Willmot -The density of the time to ruin in the classical Poisson risk model, Astin Bull, vol.35, issue.1, pp.45-60, 2005.

N. E. Karoui and &. , Chaleyat-Maurel -Un problème de réflexion et ses applications au temps local et aux équations différentielles stochastiques sur Ê, cas continu, Société mathématique de France, Astérisque, pp.117-144, 1978.

E. Egami, Yamazaki -Phase-type fitting of scale functions for spectrally negative levy processes, 2012.

A. Frolova, Y. Kabanov, and &. S. Pergamenshchikov, the insurance business risky investments are dangerous, vol.6, pp.227-235, 2002.

F. R. Gantmacher, Théorie des matrices. Tome 2 : Questions spéciales et applications, Traduit du Russe par Ch. Sarthou. Collection Universitaire de Mathé-matiques, 1966.

L. Gong and A. L. Badescu-&-e, Cheung -Recursive methods for a multidimensional risk process with common shocks, Insurance Math. Econom, vol.50, issue.1, pp.109-120, 2012.

O. Gaudoin-&-l, Doyen -Modelling and assessment of aging and efficiency of corrective and planned preventive maintenance, IEEE Trans. on Reliability, vol.60, issue.4, pp.759-769, 2011.

J. Garrido and &. Morales, On the expected discounted penalty function for Lévy risk processes, N. Am. Actuar. J, vol.10, issue.4, pp.196-218, 2006.

B. Gaujal-&-l, Rabehasaina -Open-loop control of stochastic fluid systems and applications, Oper. Res. Lett, vol.35, issue.4, pp.455-462, 2007.

F. Guillemin, Sericola -Stationary analysis of a fluid queue driven by some countable state space Markov chain, Methodol. Comput. Appl. Probab, vol.9, issue.4, pp.521-540, 2007.

H. U. Gerber and &. H. Yang, Absolute ruin probabilities in a jump diffusion risk model with investment, N. Am. Actuar. J, vol.11, issue.3, pp.159-169, 2007.

E. , Hashorva -Asymptotics and bounds for multivariate Gaussian tails, J. Theoret. Probab, vol.18, issue.1, pp.79-97, 2005.

Z. Hu and &. Jiang, On joint ruin probabilities of a two-dimensional risk model with constant interest rate, J. Appl. Probab, vol.50, issue.2, pp.309-322, 2013.

F. Hubalek and &. E. Kyprianou, Old and new examples of scale functions for spectrally negative Lévy processes, Seminar on Stochastic Analysis, Random Fields and Applications VI, vol.63, pp.119-145, 2011.

J. Haddad and R. R. Mazumdar-&-f, Piera -Pathwise comparison results for stochastic fluid networks, Queueing Syst, vol.66, issue.2, pp.155-168, 2010.

J. Ivanovs and &. Palmowski, Occupation densities in solving exit problems for Markov additive processes and their reflections, Stochastic Process. Appl, vol.122, issue.9, pp.3342-3360, 2012.

J. , Ivanovs -Markov-modulated Brownian motion with two reflecting barriers, J. Appl. Probab, vol.47, issue.4, pp.1034-1047, 2010.

R. L. Karandikar-&-v and . Kulkarni, Second-order fluid flow models : Reflected brownian motion in a random environment, Operations Research, vol.43, issue.1, pp.77-88, 1995.

A. Kuznetsov, A. E. Kyprianou, and &. Rivero, The theory of scale functions for spectrally negative lévy processes, Lévy Matters II, Lecture Notes in Mathematics, pp.97-186, 2013.

A. E. Kyprianou and &. Palmowski, A martingale review of some fluctuation theory for spectrally negative Lévy processes, Séminaire de Probabilités XXXVIII, vol.1857, pp.16-29, 2005.

A. E. Kyprianou, J. C. Pardo, and &. Rivero, Exact and asymptotic n-tuple laws at first and last passage, Ann. Appl. Probab, vol.20, issue.2, pp.522-564, 2010.

O. Kella-&-w, Stadje -Markov modulated linear fluid networks with markov additive input, J. Appl. Probab, vol.39, issue.2, pp.413-420, 2002.

F. P. Kelly-&-r and . Williams, Stochastic networks, The IMA Volumes in Mathematics and its Applications, vol.71, 1995.

O. Kella and &. Whitt, Stability and structural properties of stochastic fluid networks, J. Appl. Probab, vol.33, issue.4, pp.1169-1180, 1996.

A. E. , Kyprianou -Introductory lectures on fluctuations of Lévy processes with applications, 2006.

R. M. , Loynes -The stability of a queue with non-independent inter-arrivals and service times, Camb. Philos, vol.58, pp.497-520, 1962.

B. , Mallein -Position of the rightmost individual in a branching random walk through an interface, 2013.

S. Mercier and &. Castro, On the modelling of imperfect repairs for a continuously monitored gamma wear process through age reduction, J. Appl. Probab, vol.50, issue.4, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00984773

Z. Michna, Self-similar processes in collective risk theory, J. Appl. Math. Stochastic Anal, vol.11, issue.4, pp.429-448, 1998.

M. Miyazawa, Markov modulated two node fluid network : tail asymptotics of the stationary distribution, announced in the 2013 INFORMS Applied probability society conference in Costa Rica, 2013.

M. Morales and &. Kuznetsov, Computing the finite-time expected discounted penalty function for a family of lévy risk processes, Scand. Actuar. J, 2013.

S. Mercier-&-h, Pham -A preventive maintenance policy for a continuously monitored system with correlated wear indicators, European J. Oper. Res, vol.222, issue.2, pp.263-272, 2012.

M. , Neuts -Matrix-geometric solutions in stochastic models, Johns Hopkins Series in the Mathematical Sciences, vol.2, 1981.

J. Paulsen, Sharp conditions for certain ruin in a risk process with stochastic return on investments, Stochastic Process. Appl, vol.75, issue.1, pp.135-148, 1998.

J. Paulsen and H. K. , Gjessing -Ruin theory with stochastic return on investments, Adv. in Appl. Probab, vol.29, issue.4, pp.965-985, 1997.

F. J. Piera and R. R. Mazumdar-&-f, Guillemin -On product-form stationary distributions for reflected diffusions with jumps in the positive orthant, Adv. in Appl. Probab, vol.37, issue.1, pp.212-228, 2005.

C. J. Park-&-w and . Padgett, Accelerated degradation models for failure based on geometric Brownian motion and gamma processes, Lifetime Data Anal, vol.11, issue.4, pp.511-527, 2005.

C. Paroissin-&-l, Rabehasaina -First and last passage times of spectrally positive lévy processes with application to reliability, Methodol. Comput. Appl. Probab, 2013.

L. Rabehasaina, Moments of a Markov-modulated, irreducible network of fluid queues, J. Appl. Probab, vol.43, issue.2, pp.510-522, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00488284

, Monotonicity properties of multi-dimensional reflected diffusions in random environment and application, Stochastic Process, Appl, vol.116, issue.2, pp.178-199, 2006.

, Risk processes with interest force in Markovian environment, Stoch. Models, vol.25, issue.4, pp.580-613, 2009.

, A Markov additive risk process in dimension 2 perturbed by a fractional Brownian motion, Stochastic Process. Appl, vol.122, issue.8, pp.2925-2960, 2012.

S. Ramasubramanian, A subsidy-surplus model and the Skorokhod problem in an orthant, Math. Oper. Res, vol.25, issue.3, pp.509-538, 2000.

, A multidimensional ruin problem, Commun. Stoch. Anal, vol.6, issue.1, pp.33-47, 2012.

L. Rabehasaina-&-c and . Chi-liang, Tsai -Ruin time and aggregate claim amount up to ruin time for the perturbed risk process, Scand. Actuar. J, issue.3, pp.186-212, 2013.

J. Ren and &. S. Li, The analysis of perturbed risk processes with markovian arrivals, preprint, 2009.

L. Rabehasaina, Sericola -Stability of second order fluid flow models in a stationary ergodic environment, Ann. Appl. Probab, vol.14, issue.3, pp.1449-1473, 2003.

, A second-order Markov-modulated fluid queue with linear service rate, J. Appl. Probab, vol.41, issue.3, pp.758-777, 2004.

B. Roynette, P. Vallois, and &. Volpi, Asymptotic behavior of the hitting time, overshoot and undershoot for some Lévy processes, ESAIM Probab. Stat, vol.12, pp.58-93, 2008.

H. Seal, Risk theory and the single server queue, Mitt. Verein Schweiz. Versich. Math, vol.72, pp.171-178, 1972.

K. Sigman-&-r.-ryan, Continuous-time stochastic recursions and duality, Advances in Applied Probability, vol.32, pp.426-445, 2000.

G. C. Taylor, Representation and explicit calculation of finite-time ruin probabilities, Scand. Actuar. J, issue.1, pp.1-18, 1978.

M. I. Taksar and &. , Markussen -Optimal dynamic reinsurance policies for large insurance portfolios, Finance Stoch, vol.7, issue.1, pp.97-121, 2003.

C. C. and -. Tsai, On the discounted distribution functions of the surplus process perturbed by diffusion, Insurance Math. Econom, vol.28, issue.3, pp.401-419, 2001.

C. C. , -. Tsai, and G. E. , Willmot -A generalized defective renewal equation for the surplus process perturbed by diffusion, Insurance Math. Econom, vol.30, issue.1, pp.51-66, 2002.

G. Wang and &. Wu, Distributions for the risk process with a stochastic return on investments, Stochastic Process. Appl, vol.95, issue.2, pp.329-341, 2001.

R. Wu, G. Wang, and &. Zhang, On a joint distribution for the risk process with constant interest force, Insurance Math. Econom, vol.36, issue.3, pp.365-374, 2005.

K. Yamada, -Diffusion approximation for open state-dependent queueing networks in the heavy traffic situation, Ann. Appl. Probab, vol.5, issue.4, pp.958-982, 1995.