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A note on a Lévy insurance risk model under periodic dividend decisions
1. | College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China |
2. | Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong, China |
In this paper, we consider a spectrally negative Lévy insurance risk process with a barrier-type dividend strategy. In contrast to the traditional barrier strategy in which dividends are payable to the shareholders immediately when the surplus process reaches a fixed level b (as long as ruin has not yet occurred), it is assumed that the insurer only makes dividend decisions at some discrete time points in the spirit of [
References:
[1] |
H. Albrecher, E. C. K. Cheung and S. Thonhauser,
Randomized observation periods for the compound Poisson risk model: Dividends, ASTIN Bulletin, 41 (2011), 645-672.
|
[2] |
H. Albrecher, E. C. K. Cheung and S. Thonhauser,
Randomized observation periods for the compound Poisson risk model: The discounted penalty function, Scandinavian Actuarial Journal, 2013 (2013), 424-452.
doi: 10.1080/03461238.2011.624686. |
[3] |
H. Albrecher and H. U. Gerber,
A note on moments of dividends, Acta Mathematicae Applicatae Sinica, Acta Mathematicae Applicatae Sinica, English Series, 27 (2011), 353-354.
doi: 10.1007/s10255-011-0074-x. |
[4] |
H. Albrecher, J. Ivanovs and X. Zhou,
Exit identities for Lévy processes observed at Poisson arrival times, Bernoulli, 22 (2016), 1364-1382.
doi: 10.3150/15-BEJ695. |
[5] |
H. Albrecher, J.-F. Renaud and X. Zhou,
A Lévy insurance risk process with tax, Journal of Applied Probability, 45 (2008), 363-375.
doi: 10.1017/S0021900200004289. |
[6] |
S. Asmussen, Applied Probability and Queues, 2nd edition, Springer-Verlag, New York, 2003. |
[7] |
S. Asmussen and H. Albrecher, Ruin Probabilities, 2nd edition, World Scientific, New Jersey, 2010.
doi: 10.1142/9789814282536. |
[8] |
S. Asmussen, F. Avram and M. Usabel,
Erlangian approximations for finite-horizon ruin probabilities, ASTIN Bulletin, 32 (2002), 267-281.
doi: 10.2143/AST.32.2.1029. |
[9] |
B. Avanzi, E. C. K. Cheung, B. Wong and J.-K. Woo,
On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency, Insurance: Mathematics and Economics, 52 (2013), 98-113.
doi: 10.1016/j.insmatheco.2012.10.008. |
[10] |
E. Biffis and M. Morales,
On a generalization of the Gerber-Shiu function to path-dependent penalties, Insurance: Mathematics and Economics, 46 (2010), 92-97.
doi: 10.1016/j.insmatheco.2009.08.011. |
[11] |
P. Carr,
Randomization and the American put, Review of Financial Studies, 11 (1998), 597-626.
|
[12] |
E. C. K. Cheung,
A generalized penalty function in Sparre Andersen risk models with surplus-dependent premium, Insurance: Mathematics and Economics, 48 (2011), 384-397.
doi: 10.1016/j.insmatheco.2011.01.006. |
[13] |
E. C. K. Cheung, D. C. M. Dickson and S. Drekic,
Moments of discounted dividends for a threshold strategy in the compound Poisson risk model, North American Actuarial Journal, 12 (2008), 299-318.
doi: 10.1080/10920277.2008.10597523. |
[14] |
I. Czarna and Z. Palmowski,
Ruin probability with Parisian delay for a spectrally negative Lévy risk process, Journal of Applied Probability, 48 (2011), 984-1002.
doi: 10.1017/S0021900200008573. |
[15] |
D. C. M. Dickson and C. Hipp,
On the time to ruin for Erlang(2) risk processes, Insurance: Mathematics and Economics, 29 (2001), 333-344.
doi: 10.1016/S0167-6687(01)00091-9. |
[16] |
D. C. M. Dickson and H. R. Waters,
Some optimal dividends problems, ASTIN Bulletin, 34 (2004), 49-74.
doi: 10.2143/AST.34.1.504954. |
[17] |
F. Dufresne and H.U. Gerber,
Risk theory for the compound Poisson process that is perturbed by diffusion, Insurance: Mathematics and Economics, 10 (1991), 51-59.
doi: 10.1016/0167-6687(91)90023-Q. |
[18] |
R. Feng,
A matrix operator approach to the analysis of ruin-related quantities in the phasetype renewal risk model, Bulletin of the Swiss Association of Actuaries, 2009 (2009), 71-87.
|
[19] |
R. Feng and Y. Shimizu,
On a generalization from ruin to default in a Lévy insurance risk model, Methodology and Computing in Applied Probability, 15 (2013), 773-802.
doi: 10.1007/s11009-012-9282-y. |
[20] |
H. Furrer,
Risk processes perturbed by α-stable Lévy motion, Scandinavian Actuarial Journal, 1998 (1998), 59-74.
doi: 10.1080/03461238.1998.10413992. |
[21] |
J. Garrido and M. Morales,
On the expected discounted penalty function for Lévy risk processes, North American Actuarial Journal, 10 (2006), 196-218.
doi: 10.1080/10920277.2006.10597421. |
[22] |
H. U. Gerber, An Introduction to Mathematical Risk Theory, Huebner Foundation Monograph 8, Richard D. Irwin: Homewood, Illinois, 1979. |
[23] |
H. U. Gerber and E. S. W. Shiu,
On optimal dividends: From reflection to refraction, Journal of Computational and Applied Mathematics, 186 (2006), 4-22.
doi: 10.1016/j.cam.2005.03.062. |
[24] |
H. U. Gerber and E. S. W. Shiu,
On the time value of ruin, North American Actuarial Journal, 2 (1998), 48-78.
doi: 10.1080/10920277.1998.10595671. |
[25] |
H. U. Gerber and E. S. W. Shiu,
Optimal dividends: Analysis with Brownian Motion, North American Actuarial Journal, 8 (2004), 1-20.
doi: 10.1080/10920277.2004.10596125. |
[26] |
M. Huzak, M. Perman, H. Šikič and Z. Vondraček,
Ruin probabilities and decompositions for general perturbed risk processes, Annals of Applied Probability, 14 (2004), 1378-1397.
doi: 10.1214/105051604000000332. |
[27] |
A. E. Kyprianou, Fluctuations of Lévy Processes with Applications: Introductory Lectures, 2nd edition, Springer-Verlag, Berlin Heidelberg, 2014.
doi: 10.1007/978-3-642-37632-0. |
[28] |
A. E. Kyprianou, Gerber-Shiu Risk Theory, Springer, Cham Heidelberg New York Dordrecht London, 2013. '
doi: 10.1007/978-3-319-02303-8. |
[29] |
A. E. Kyprianou and R. L. Loeffen,
Refracted Lévy processes, Annales de l'Institut Henri Poincaré -Probabilités et Statistiques, 46 (2010), 24-44.
doi: 10.1214/08-AIHP307. |
[30] |
A. E. Kyprianou and Z. Palmowski,
Distributional study of De Finetti's dividend problem for a general Lévy insurance risk process, Journal of Applied Probability, 44 (2007), 428-443.
doi: 10.1017/S0021900200117930. |
[31] |
A. E. Kyprianou and Z. Palmowski,
Fluctuations of spectrally negative Markov additive process, Séminaire de Probabilitiés XLI, Lecture Notes in Mathematics, 1934 (2008), 121-135.
doi: 10.1007/978-3-540-77913-1_5. |
[32] |
A. E. Kyprianou and M. R. Pistorius,
Perpetual options and Canadization through fluctuation theory, Annals of Applied Probability, 13 (2003), 1077-1098.
doi: 10.1214/aoap/1060202835. |
[33] |
A. E. Kyprianou and X. Zhou,
General tax structures and the Lévy insurance risk model, Journal of Applied Probability, 46 (2009), 1146-1156.
doi: 10.1017/S0021900200006197. |
[34] |
X. S. Lin, G. E. Willmot and S. Drekic,
The compound Poisson risk model with a constant dividend barrier: Analysis of the Gerber-Shiu discounted penalty function, Insurance: Mathematics and Economics, 33 (2003), 551-566.
doi: 10.1016/j.insmatheco.2003.08.004. |
[35] |
B. G. Lindsay, R. S. Pilla and P. Basak,
Moment-based approximations of distributions using mixtures: Theory and applications, Annals of the Institute of Statistical Mathematics, 52 (2000), 215-230.
doi: 10.1023/A:1004105603806. |
[36] |
R. Loeffen, I. Czarna and Z. Palmowski,
Parisian ruin probability for spectrally negative Lévy processes, Bernoulli, 19 (2013), 599-609.
doi: 10.3150/11-BEJ404. |
[37] |
J.-F. Renaud and X. Zhou,
Distribution of the present value of dividend payments in a Lévy risk model, Journal of Applied Probability, 44 (2007), 420-427.
doi: 10.1017/S0021900200117929. |
[38] |
V. Ramaswami, D. G. Woolford and D. A. Stanford,
The Erlangization method for Markovian fluid flows, Annals of Operations Research, 160 (2008), 215-225.
doi: 10.1007/s10479-008-0309-2. |
[39] |
Z. B. Salah and M. Morales,
Lévy systems and the time value of ruin for Markov additive processes, European Actuarial Journal, 2 (2012), 289-317.
doi: 10.1007/s13385-012-0053-5. |
[40] |
H. Schmidli,
Distribution of the first ladder height of a stationary risk process perturbed by α-stable Lévy motion, Insurance: Mathematics and Economics, 28 (2001), 13-20.
doi: 10.1016/S0167-6687(00)00062-7. |
[41] |
D. A. Stanford, F. Avram, A. L. Badescu, L. Breuer, A. Da Silva Soares and G. Latouche,
Phase-type approximations to finite-time ruin probabilities in the Sparre-Anderson and stationary renewal risk models, ASTIN Bulletin, 35 (2005), 131-144.
doi: 10.2143/AST.35.1.583169. |
[42] |
D. A. Stanford, K. Yu and J. Ren,
Erlangian approximation to finite time ruin probabilities in perturbed risk models, Scandinavian Actuarial Journal, 2011 (2011), 38-58.
doi: 10.1080/03461230903421492. |
[43] |
C. C.-L. Tsai and G. E. Willmot,
A generalized defective renewal equation for the surplus process perturbed by diffusion, Insurance: Mathematics and Economics, 30 (2002), 51-66.
doi: 10.1016/S0167-6687(01)00096-8. |
[44] |
Z. Zhang,
On a risk model with randomized dividend-decision times, Journal of Industrial and Management Optimization, 10 (2014), 1041-1058.
doi: 10.3934/jimo.2014.10.1041. |
[45] |
Z. Zhang and E. C. K. Cheung,
The Markov additive risk process under an Erlangized dividend barrier strategy, Methodology and Computing in Applied Probability, 18 (2016), 275-306.
doi: 10.1007/s11009-014-9414-7. |
[46] |
Z. Zhang, C. K. Cheung and H. Yang,
Lévy insurance risk process with Poissonian taxation, Scandinavian Actuarial Journal, 2017 (2017), 51-87.
doi: 10.1080/03461238.2015.1062042. |
[47] |
Z. Zhang, C. Liu and Y. Yang,
On a perturbed compound Poisson model with varying premium rates, Journal of Industrial and Management Optimization, 13 (2017), 721-736.
doi: 10.3934/jimo.2016043. |
show all references
References:
[1] |
H. Albrecher, E. C. K. Cheung and S. Thonhauser,
Randomized observation periods for the compound Poisson risk model: Dividends, ASTIN Bulletin, 41 (2011), 645-672.
|
[2] |
H. Albrecher, E. C. K. Cheung and S. Thonhauser,
Randomized observation periods for the compound Poisson risk model: The discounted penalty function, Scandinavian Actuarial Journal, 2013 (2013), 424-452.
doi: 10.1080/03461238.2011.624686. |
[3] |
H. Albrecher and H. U. Gerber,
A note on moments of dividends, Acta Mathematicae Applicatae Sinica, Acta Mathematicae Applicatae Sinica, English Series, 27 (2011), 353-354.
doi: 10.1007/s10255-011-0074-x. |
[4] |
H. Albrecher, J. Ivanovs and X. Zhou,
Exit identities for Lévy processes observed at Poisson arrival times, Bernoulli, 22 (2016), 1364-1382.
doi: 10.3150/15-BEJ695. |
[5] |
H. Albrecher, J.-F. Renaud and X. Zhou,
A Lévy insurance risk process with tax, Journal of Applied Probability, 45 (2008), 363-375.
doi: 10.1017/S0021900200004289. |
[6] |
S. Asmussen, Applied Probability and Queues, 2nd edition, Springer-Verlag, New York, 2003. |
[7] |
S. Asmussen and H. Albrecher, Ruin Probabilities, 2nd edition, World Scientific, New Jersey, 2010.
doi: 10.1142/9789814282536. |
[8] |
S. Asmussen, F. Avram and M. Usabel,
Erlangian approximations for finite-horizon ruin probabilities, ASTIN Bulletin, 32 (2002), 267-281.
doi: 10.2143/AST.32.2.1029. |
[9] |
B. Avanzi, E. C. K. Cheung, B. Wong and J.-K. Woo,
On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency, Insurance: Mathematics and Economics, 52 (2013), 98-113.
doi: 10.1016/j.insmatheco.2012.10.008. |
[10] |
E. Biffis and M. Morales,
On a generalization of the Gerber-Shiu function to path-dependent penalties, Insurance: Mathematics and Economics, 46 (2010), 92-97.
doi: 10.1016/j.insmatheco.2009.08.011. |
[11] |
P. Carr,
Randomization and the American put, Review of Financial Studies, 11 (1998), 597-626.
|
[12] |
E. C. K. Cheung,
A generalized penalty function in Sparre Andersen risk models with surplus-dependent premium, Insurance: Mathematics and Economics, 48 (2011), 384-397.
doi: 10.1016/j.insmatheco.2011.01.006. |
[13] |
E. C. K. Cheung, D. C. M. Dickson and S. Drekic,
Moments of discounted dividends for a threshold strategy in the compound Poisson risk model, North American Actuarial Journal, 12 (2008), 299-318.
doi: 10.1080/10920277.2008.10597523. |
[14] |
I. Czarna and Z. Palmowski,
Ruin probability with Parisian delay for a spectrally negative Lévy risk process, Journal of Applied Probability, 48 (2011), 984-1002.
doi: 10.1017/S0021900200008573. |
[15] |
D. C. M. Dickson and C. Hipp,
On the time to ruin for Erlang(2) risk processes, Insurance: Mathematics and Economics, 29 (2001), 333-344.
doi: 10.1016/S0167-6687(01)00091-9. |
[16] |
D. C. M. Dickson and H. R. Waters,
Some optimal dividends problems, ASTIN Bulletin, 34 (2004), 49-74.
doi: 10.2143/AST.34.1.504954. |
[17] |
F. Dufresne and H.U. Gerber,
Risk theory for the compound Poisson process that is perturbed by diffusion, Insurance: Mathematics and Economics, 10 (1991), 51-59.
doi: 10.1016/0167-6687(91)90023-Q. |
[18] |
R. Feng,
A matrix operator approach to the analysis of ruin-related quantities in the phasetype renewal risk model, Bulletin of the Swiss Association of Actuaries, 2009 (2009), 71-87.
|
[19] |
R. Feng and Y. Shimizu,
On a generalization from ruin to default in a Lévy insurance risk model, Methodology and Computing in Applied Probability, 15 (2013), 773-802.
doi: 10.1007/s11009-012-9282-y. |
[20] |
H. Furrer,
Risk processes perturbed by α-stable Lévy motion, Scandinavian Actuarial Journal, 1998 (1998), 59-74.
doi: 10.1080/03461238.1998.10413992. |
[21] |
J. Garrido and M. Morales,
On the expected discounted penalty function for Lévy risk processes, North American Actuarial Journal, 10 (2006), 196-218.
doi: 10.1080/10920277.2006.10597421. |
[22] |
H. U. Gerber, An Introduction to Mathematical Risk Theory, Huebner Foundation Monograph 8, Richard D. Irwin: Homewood, Illinois, 1979. |
[23] |
H. U. Gerber and E. S. W. Shiu,
On optimal dividends: From reflection to refraction, Journal of Computational and Applied Mathematics, 186 (2006), 4-22.
doi: 10.1016/j.cam.2005.03.062. |
[24] |
H. U. Gerber and E. S. W. Shiu,
On the time value of ruin, North American Actuarial Journal, 2 (1998), 48-78.
doi: 10.1080/10920277.1998.10595671. |
[25] |
H. U. Gerber and E. S. W. Shiu,
Optimal dividends: Analysis with Brownian Motion, North American Actuarial Journal, 8 (2004), 1-20.
doi: 10.1080/10920277.2004.10596125. |
[26] |
M. Huzak, M. Perman, H. Šikič and Z. Vondraček,
Ruin probabilities and decompositions for general perturbed risk processes, Annals of Applied Probability, 14 (2004), 1378-1397.
doi: 10.1214/105051604000000332. |
[27] |
A. E. Kyprianou, Fluctuations of Lévy Processes with Applications: Introductory Lectures, 2nd edition, Springer-Verlag, Berlin Heidelberg, 2014.
doi: 10.1007/978-3-642-37632-0. |
[28] |
A. E. Kyprianou, Gerber-Shiu Risk Theory, Springer, Cham Heidelberg New York Dordrecht London, 2013. '
doi: 10.1007/978-3-319-02303-8. |
[29] |
A. E. Kyprianou and R. L. Loeffen,
Refracted Lévy processes, Annales de l'Institut Henri Poincaré -Probabilités et Statistiques, 46 (2010), 24-44.
doi: 10.1214/08-AIHP307. |
[30] |
A. E. Kyprianou and Z. Palmowski,
Distributional study of De Finetti's dividend problem for a general Lévy insurance risk process, Journal of Applied Probability, 44 (2007), 428-443.
doi: 10.1017/S0021900200117930. |
[31] |
A. E. Kyprianou and Z. Palmowski,
Fluctuations of spectrally negative Markov additive process, Séminaire de Probabilitiés XLI, Lecture Notes in Mathematics, 1934 (2008), 121-135.
doi: 10.1007/978-3-540-77913-1_5. |
[32] |
A. E. Kyprianou and M. R. Pistorius,
Perpetual options and Canadization through fluctuation theory, Annals of Applied Probability, 13 (2003), 1077-1098.
doi: 10.1214/aoap/1060202835. |
[33] |
A. E. Kyprianou and X. Zhou,
General tax structures and the Lévy insurance risk model, Journal of Applied Probability, 46 (2009), 1146-1156.
doi: 10.1017/S0021900200006197. |
[34] |
X. S. Lin, G. E. Willmot and S. Drekic,
The compound Poisson risk model with a constant dividend barrier: Analysis of the Gerber-Shiu discounted penalty function, Insurance: Mathematics and Economics, 33 (2003), 551-566.
doi: 10.1016/j.insmatheco.2003.08.004. |
[35] |
B. G. Lindsay, R. S. Pilla and P. Basak,
Moment-based approximations of distributions using mixtures: Theory and applications, Annals of the Institute of Statistical Mathematics, 52 (2000), 215-230.
doi: 10.1023/A:1004105603806. |
[36] |
R. Loeffen, I. Czarna and Z. Palmowski,
Parisian ruin probability for spectrally negative Lévy processes, Bernoulli, 19 (2013), 599-609.
doi: 10.3150/11-BEJ404. |
[37] |
J.-F. Renaud and X. Zhou,
Distribution of the present value of dividend payments in a Lévy risk model, Journal of Applied Probability, 44 (2007), 420-427.
doi: 10.1017/S0021900200117929. |
[38] |
V. Ramaswami, D. G. Woolford and D. A. Stanford,
The Erlangization method for Markovian fluid flows, Annals of Operations Research, 160 (2008), 215-225.
doi: 10.1007/s10479-008-0309-2. |
[39] |
Z. B. Salah and M. Morales,
Lévy systems and the time value of ruin for Markov additive processes, European Actuarial Journal, 2 (2012), 289-317.
doi: 10.1007/s13385-012-0053-5. |
[40] |
H. Schmidli,
Distribution of the first ladder height of a stationary risk process perturbed by α-stable Lévy motion, Insurance: Mathematics and Economics, 28 (2001), 13-20.
doi: 10.1016/S0167-6687(00)00062-7. |
[41] |
D. A. Stanford, F. Avram, A. L. Badescu, L. Breuer, A. Da Silva Soares and G. Latouche,
Phase-type approximations to finite-time ruin probabilities in the Sparre-Anderson and stationary renewal risk models, ASTIN Bulletin, 35 (2005), 131-144.
doi: 10.2143/AST.35.1.583169. |
[42] |
D. A. Stanford, K. Yu and J. Ren,
Erlangian approximation to finite time ruin probabilities in perturbed risk models, Scandinavian Actuarial Journal, 2011 (2011), 38-58.
doi: 10.1080/03461230903421492. |
[43] |
C. C.-L. Tsai and G. E. Willmot,
A generalized defective renewal equation for the surplus process perturbed by diffusion, Insurance: Mathematics and Economics, 30 (2002), 51-66.
doi: 10.1016/S0167-6687(01)00096-8. |
[44] |
Z. Zhang,
On a risk model with randomized dividend-decision times, Journal of Industrial and Management Optimization, 10 (2014), 1041-1058.
doi: 10.3934/jimo.2014.10.1041. |
[45] |
Z. Zhang and E. C. K. Cheung,
The Markov additive risk process under an Erlangized dividend barrier strategy, Methodology and Computing in Applied Probability, 18 (2016), 275-306.
doi: 10.1007/s11009-014-9414-7. |
[46] |
Z. Zhang, C. K. Cheung and H. Yang,
Lévy insurance risk process with Poissonian taxation, Scandinavian Actuarial Journal, 2017 (2017), 51-87.
doi: 10.1080/03461238.2015.1062042. |
[47] |
Z. Zhang, C. Liu and Y. Yang,
On a perturbed compound Poisson model with varying premium rates, Journal of Industrial and Management Optimization, 13 (2017), 721-736.
doi: 10.3934/jimo.2016043. |



1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
| 10.698 | 10.816 | 10.858 | 10.879 | 10.892 | 10.900 | 10.906 | 10.911 |
| 39.9857 | 40.1571 | 40.2133 | 40.2412 | 40.2579 | 40.2690 | 40.2769 | 40.2828 |
| 15.9415 | 16.0128 | 16.0385 | 16.0478 | 16.0577 | 16.0627 | 16.0647 | 16.0660 |
| 47.0084 | 47.2091 | 47.2758 | 47.3082 | 47.3283 | 47.3419 | 47.3508 | 47.3574 |
| 12.1893 | 12.2434 | 12.2672 | 12.2727 | 12.2828 | 12.2921 | 12.2919 | 12.2908 |
| 47.7133 | 48.0345 | 48.1452 | 48.1979 | 48.2329 | 48.2543 | 48.2688 | 48.2802 |
| 12.1021 | 12.1415 | 12.1638 | 12.1622 | 12.1763 | 12.1869 | 12.1830 | 12.1789 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
| 10.698 | 10.816 | 10.858 | 10.879 | 10.892 | 10.900 | 10.906 | 10.911 |
| 39.9857 | 40.1571 | 40.2133 | 40.2412 | 40.2579 | 40.2690 | 40.2769 | 40.2828 |
| 15.9415 | 16.0128 | 16.0385 | 16.0478 | 16.0577 | 16.0627 | 16.0647 | 16.0660 |
| 47.0084 | 47.2091 | 47.2758 | 47.3082 | 47.3283 | 47.3419 | 47.3508 | 47.3574 |
| 12.1893 | 12.2434 | 12.2672 | 12.2727 | 12.2828 | 12.2921 | 12.2919 | 12.2908 |
| 47.7133 | 48.0345 | 48.1452 | 48.1979 | 48.2329 | 48.2543 | 48.2688 | 48.2802 |
| 12.1021 | 12.1415 | 12.1638 | 12.1622 | 12.1763 | 12.1869 | 12.1830 | 12.1789 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
| 13.036 | 13.209 | 13.270 | 13.300 | 13.319 | 13.332 | 13.339 | 13.349 |
| 13.1645 | 13.2114 | 13.2266 | 13.2341 | 13.2386 | 13.2416 | 13.2437 | 13.2454 |
| 19.6093 | 19.6802 | 19.7034 | 19.7146 | 19.7215 | 19.7263 | 19.7293 | 19.7314 |
| 36.2764 | 36.4057 | 36.4477 | 36.4684 | 36.4808 | 36.4890 | 36.4948 | 36.4993 |
| 17.2879 | 17.3529 | 17.3748 | 17.3846 | 17.3911 | 17.3961 | 17.3986 | 17.3996 |
| 43.8083 | 43.9645 | 44.0151 | 44.0402 | 44.0551 | 44.0650 | 44.0721 | 44.0775 |
| 13.5952 | 13.6492 | 13.6681 | 13.6758 | 13.6813 | 13.6864 | 13.6879 | 13.6877 |
| 46.9747 | 47.3146 | 47.4309 | 47.4872 | 47.5227 | 47.5474 | 47.5610 | 47.5771 |
| 13.0272 | 13.0611 | 13.0776 | 13.0796 | 13.0908 | 13.0968 | 13.0962 | 13.0907 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
| 13.036 | 13.209 | 13.270 | 13.300 | 13.319 | 13.332 | 13.339 | 13.349 |
| 13.1645 | 13.2114 | 13.2266 | 13.2341 | 13.2386 | 13.2416 | 13.2437 | 13.2454 |
| 19.6093 | 19.6802 | 19.7034 | 19.7146 | 19.7215 | 19.7263 | 19.7293 | 19.7314 |
| 36.2764 | 36.4057 | 36.4477 | 36.4684 | 36.4808 | 36.4890 | 36.4948 | 36.4993 |
| 17.2879 | 17.3529 | 17.3748 | 17.3846 | 17.3911 | 17.3961 | 17.3986 | 17.3996 |
| 43.8083 | 43.9645 | 44.0151 | 44.0402 | 44.0551 | 44.0650 | 44.0721 | 44.0775 |
| 13.5952 | 13.6492 | 13.6681 | 13.6758 | 13.6813 | 13.6864 | 13.6879 | 13.6877 |
| 46.9747 | 47.3146 | 47.4309 | 47.4872 | 47.5227 | 47.5474 | 47.5610 | 47.5771 |
| 13.0272 | 13.0611 | 13.0776 | 13.0796 | 13.0908 | 13.0968 | 13.0962 | 13.0907 |
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