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January  2018, 14(1): 35-63. doi: 10.3934/jimo.2017036

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

* Corresponding author: Zhimin Zhang

Received  September 2015 Revised  February 2017 Published  April 2017

Fund Project: Zhimin Zhang is supported by the National Natural Science Foundation of China [11471058,11661074], the Natural Science Foundation Project of CQ CSTC of China [cstc2014jcyjA00007] and MOE (Ministry of Education in China) Project of Humanities and Social Sciences (Project No. 16YJC910005). Eric Cheung gratefully acknowledges the support from the Research Grants Council of the Hong Kong Special Administrative Region (Project Number: HKU 17324016). This research is also partially supported by the CAE 2013 research grant from the Society of Actuaries. Any opinions, finding, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the SOA

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 [1]. Under such a dividend strategy with Erlang inter-dividend-decision times, expressions for the Gerber-Shiu expected discounted penalty function proposed in [24] and the moments of total discounted dividends payable until ruin are derived. The results are expressed in terms of the scale functions of a spectrally negative Lévy process and an embedded spectrally negative Markov additive process. Our analyses rely on the introduction of a potential measure associated with an Erlang random variable. Numerical illustrations are also given.

Citation: Zhimin Zhang, Eric C. K. Cheung. A note on a Lévy insurance risk model under periodic dividend decisions. Journal of Industrial & Management Optimization, 2018, 14 (1) : 35-63. doi: 10.3934/jimo.2017036
References:
[1]

H. AlbrecherE. C. K. Cheung and S. Thonhauser, Randomized observation periods for the compound Poisson risk model: Dividends, ASTIN Bulletin, 41 (2011), 645-672. Google Scholar

[2]

H. AlbrecherE. 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. Google Scholar

[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. Google Scholar

[4]

H. AlbrecherJ. Ivanovs and X. Zhou, Exit identities for Lévy processes observed at Poisson arrival times, Bernoulli, 22 (2016), 1364-1382. doi: 10.3150/15-BEJ695. Google Scholar

[5]

H. AlbrecherJ.-F. Renaud and X. Zhou, A Lévy insurance risk process with tax, Journal of Applied Probability, 45 (2008), 363-375. doi: 10.1017/S0021900200004289. Google Scholar

[6]

S. Asmussen, Applied Probability and Queues, 2nd edition, Springer-Verlag, New York, 2003. Google Scholar

[7]

S. Asmussen and H. Albrecher, Ruin Probabilities, 2nd edition, World Scientific, New Jersey, 2010. doi: 10.1142/9789814282536. Google Scholar

[8]

S. AsmussenF. Avram and M. Usabel, Erlangian approximations for finite-horizon ruin probabilities, ASTIN Bulletin, 32 (2002), 267-281. doi: 10.2143/AST.32.2.1029. Google Scholar

[9]

B. AvanziE. C. K. CheungB. 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. Google Scholar

[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. Google Scholar

[11]

P. Carr, Randomization and the American put, Review of Financial Studies, 11 (1998), 597-626. Google Scholar

[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. Google Scholar

[13]

E. C. K. CheungD. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

H. Furrer, Risk processes perturbed by α-stable Lévy motion, Scandinavian Actuarial Journal, 1998 (1998), 59-74. doi: 10.1080/03461238.1998.10413992. Google Scholar

[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. Google Scholar

[22]

H. U. Gerber, An Introduction to Mathematical Risk Theory, Huebner Foundation Monograph 8, Richard D. Irwin: Homewood, Illinois, 1979. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

M. HuzakM. PermanH. 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. Google Scholar

[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. Google Scholar

[28]

A. E. Kyprianou, Gerber-Shiu Risk Theory, Springer, Cham Heidelberg New York Dordrecht London, 2013. ' doi: 10.1007/978-3-319-02303-8. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[34]

X. S. LinG. 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. Google Scholar

[35]

B. G. LindsayR. 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. Google Scholar

[36]

R. LoeffenI. Czarna and Z. Palmowski, Parisian ruin probability for spectrally negative Lévy processes, Bernoulli, 19 (2013), 599-609. doi: 10.3150/11-BEJ404. Google Scholar

[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. Google Scholar

[38]

V. RamaswamiD. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[41]

D. A. StanfordF. AvramA. L. BadescuL. BreuerA. 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. Google Scholar

[42]

D. A. StanfordK. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[46]

Z. ZhangC. 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. Google Scholar

[47]

Z. ZhangC. 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. Google Scholar

show all references

References:
[1]

H. AlbrecherE. C. K. Cheung and S. Thonhauser, Randomized observation periods for the compound Poisson risk model: Dividends, ASTIN Bulletin, 41 (2011), 645-672. Google Scholar

[2]

H. AlbrecherE. 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. Google Scholar

[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. Google Scholar

[4]

H. AlbrecherJ. Ivanovs and X. Zhou, Exit identities for Lévy processes observed at Poisson arrival times, Bernoulli, 22 (2016), 1364-1382. doi: 10.3150/15-BEJ695. Google Scholar

[5]

H. AlbrecherJ.-F. Renaud and X. Zhou, A Lévy insurance risk process with tax, Journal of Applied Probability, 45 (2008), 363-375. doi: 10.1017/S0021900200004289. Google Scholar

[6]

S. Asmussen, Applied Probability and Queues, 2nd edition, Springer-Verlag, New York, 2003. Google Scholar

[7]

S. Asmussen and H. Albrecher, Ruin Probabilities, 2nd edition, World Scientific, New Jersey, 2010. doi: 10.1142/9789814282536. Google Scholar

[8]

S. AsmussenF. Avram and M. Usabel, Erlangian approximations for finite-horizon ruin probabilities, ASTIN Bulletin, 32 (2002), 267-281. doi: 10.2143/AST.32.2.1029. Google Scholar

[9]

B. AvanziE. C. K. CheungB. 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. Google Scholar

[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. Google Scholar

[11]

P. Carr, Randomization and the American put, Review of Financial Studies, 11 (1998), 597-626. Google Scholar

[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. Google Scholar

[13]

E. C. K. CheungD. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

H. Furrer, Risk processes perturbed by α-stable Lévy motion, Scandinavian Actuarial Journal, 1998 (1998), 59-74. doi: 10.1080/03461238.1998.10413992. Google Scholar

[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. Google Scholar

[22]

H. U. Gerber, An Introduction to Mathematical Risk Theory, Huebner Foundation Monograph 8, Richard D. Irwin: Homewood, Illinois, 1979. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

M. HuzakM. PermanH. 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. Google Scholar

[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. Google Scholar

[28]

A. E. Kyprianou, Gerber-Shiu Risk Theory, Springer, Cham Heidelberg New York Dordrecht London, 2013. ' doi: 10.1007/978-3-319-02303-8. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[34]

X. S. LinG. 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. Google Scholar

[35]

B. G. LindsayR. 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. Google Scholar

[36]

R. LoeffenI. Czarna and Z. Palmowski, Parisian ruin probability for spectrally negative Lévy processes, Bernoulli, 19 (2013), 599-609. doi: 10.3150/11-BEJ404. Google Scholar

[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. Google Scholar

[38]

V. RamaswamiD. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[41]

D. A. StanfordF. AvramA. L. BadescuL. BreuerA. 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. Google Scholar

[42]

D. A. StanfordK. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[46]

Z. ZhangC. 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. Google Scholar

[47]

Z. ZhangC. 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. Google Scholar

Figure 1.  Impact of the parameter $\beta$. (a) Laplace transform of ruin time $\phi_\delta(u;b)$. (b) Expected discounted dividends $V_{1, \delta}(u;b)$
Figure 2.  Impact of claim distributions. (a) Laplace transform of ruin time $\phi_\delta(u;b)$. (b) Expected discounted dividends $V_{1, \delta}(u;b)$
Figure 3.  Expected discounted dividends $V_{1, \delta}(u;b)$ as a function of $b$. (a) Brownian motion model. (b) Compound Poisson model with exponential claims
Table 1.  Exact values of $b^*$, $V_{1, \delta}(u;b^*)$ and $SD_\delta(u;b^*)$ in the Brownian motion model
$ m$12345678
$b^{*}$10.69810.81610.85810.87910.89210.90010.90610.911
$V_{1, \delta}(5;b^{*})$39.985740.157140.213340.241240.257940.269040.276940.2828
$SD_\delta(5;b^{*})$15.941516.012816.038516.047816.057716.062716.064716.0660
$V_{1, \delta}(10;b^{*})$47.008447.209147.275847.308247.328347.341947.350847.3574
$SD_\delta(10;b^{*})$12.189312.243412.267212.272712.282812.292112.291912.2908
$V_{1, \delta}(b^{*};b^{*})$47.713348.034548.145248.197948.232948.254348.268848.2802
$SD_\delta(b^{*};b^{*})$12.102112.141512.163812.162212.176312.186912.183012.1789
$ m$12345678
$b^{*}$10.69810.81610.85810.87910.89210.90010.90610.911
$V_{1, \delta}(5;b^{*})$39.985740.157140.213340.241240.257940.269040.276940.2828
$SD_\delta(5;b^{*})$15.941516.012816.038516.047816.057716.062716.064716.0660
$V_{1, \delta}(10;b^{*})$47.008447.209147.275847.308247.328347.341947.350847.3574
$SD_\delta(10;b^{*})$12.189312.243412.267212.272712.282812.292112.291912.2908
$V_{1, \delta}(b^{*};b^{*})$47.713348.034548.145248.197948.232948.254348.268848.2802
$SD_\delta(b^{*};b^{*})$12.102112.141512.163812.162212.176312.186912.183012.1789
Table 2.  Exact values of $b^*$, $V_{1, \delta}(u;b^*)$ and $SD_\delta(u;b^*)$ in the compound Poisson model (ⅱ)
$m$12345678
$b^{*}$13.03613.20913.27013.30013.31913.33213.33913.349
$V_{1, \delta}(0;b^{*})$13.164513.211413.226613.234113.238613.241613.243713.2454
$SD_\delta(0;b^{*})$19.609319.680219.703419.714619.721519.726319.729319.7314
$V_{1, \delta}(5;b^{*})$36.276436.405736.447736.468436.480836.489036.494836.4993
$SD_\delta(5;b^{*})$17.287917.352917.374817.384617.391117.396117.398617.3996
$V_{1, \delta}(10;b^{*})$43.808343.964544.015144.040244.055144.065044.072144.0775
$SD_\delta(10;b^{*})$13.595213.649213.668113.675813.681313.686413.687913.6877
$V_{1, \delta}(b^{*};b^{*})$46.974747.314647.430947.487247.522747.547447.561047.5771
$SD_\delta(b^{*};b^{*})$13.027213.061113.077613.079613.090813.096813.096213.0907
$m$12345678
$b^{*}$13.03613.20913.27013.30013.31913.33213.33913.349
$V_{1, \delta}(0;b^{*})$13.164513.211413.226613.234113.238613.241613.243713.2454
$SD_\delta(0;b^{*})$19.609319.680219.703419.714619.721519.726319.729319.7314
$V_{1, \delta}(5;b^{*})$36.276436.405736.447736.468436.480836.489036.494836.4993
$SD_\delta(5;b^{*})$17.287917.352917.374817.384617.391117.396117.398617.3996
$V_{1, \delta}(10;b^{*})$43.808343.964544.015144.040244.055144.065044.072144.0775
$SD_\delta(10;b^{*})$13.595213.649213.668113.675813.681313.686413.687913.6877
$V_{1, \delta}(b^{*};b^{*})$46.974747.314647.430947.487247.522747.547447.561047.5771
$SD_\delta(b^{*};b^{*})$13.027213.061113.077613.079613.090813.096813.096213.0907
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