• Previous Article
    $\mathcal{H}_∞$ filtering for switched nonlinear systems: A state projection method
  • JIMO Home
  • This Issue
  • Next Article
    Gap functions and Hausdorff continuity of solution mappings to parametric strong vector quasiequilibrium problems
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.

[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.

[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. 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.

[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.

[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. 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.

[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.

[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. 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.

[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. 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.

[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. 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.

[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.

[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.

[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. 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.

[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. 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.

[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.

[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. 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.

[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.

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.

[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.

[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. 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.

[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.

[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. 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.

[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.

[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. 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.

[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. 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.

[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. 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.

[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.

[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.

[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. 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.

[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. 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.

[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.

[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. 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.

[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.

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
[1]

Yongxia Zhao, Rongming Wang, Chuancun Yin. Optimal dividends and capital injections for a spectrally positive Lévy process. Journal of Industrial & Management Optimization, 2017, 13 (1) : 1-21. doi: 10.3934/jimo.2016001

[2]

Steve Drekic, Jae-Kyung Woo, Ran Xu. A threshold-based risk process with a waiting period to pay dividends. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1179-1201. doi: 10.3934/jimo.2018005

[3]

Jiangyan Peng, Dingcheng Wang. Asymptotics for ruin probabilities of a non-standard renewal risk model with dependence structures and exponential Lévy process investment returns. Journal of Industrial & Management Optimization, 2017, 13 (1) : 155-185. doi: 10.3934/jimo.2016010

[4]

Yong-Kum Cho. On the Boltzmann equation with the symmetric stable Lévy process. Kinetic & Related Models, 2015, 8 (1) : 53-77. doi: 10.3934/krm.2015.8.53

[5]

Yang Yang, Kaiyong Wang, Jiajun Liu, Zhimin Zhang. Asymptotics for a bidimensional risk model with two geometric Lévy price processes. Journal of Industrial & Management Optimization, 2018, 13 (5) : 1-25. doi: 10.3934/jimo.2018053

[6]

Hongjun Gao, Fei Liang. On the stochastic beam equation driven by a Non-Gaussian Lévy process. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1027-1045. doi: 10.3934/dcdsb.2014.19.1027

[7]

Yang Yang, Kam C. Yuen, Jun-Feng Liu. Asymptotics for ruin probabilities in Lévy-driven risk models with heavy-tailed claims. Journal of Industrial & Management Optimization, 2018, 14 (1) : 231-247. doi: 10.3934/jimo.2017044

[8]

Manman Li, George Yin. Optimal threshold strategies with capital injections in a spectrally negative Lévy risk model. Journal of Industrial & Management Optimization, 2018, 13 (5) : 1-19. doi: 10.3934/jimo.2018055

[9]

Wen Chen, Song Wang. A finite difference method for pricing European and American options under a geometric Lévy process. Journal of Industrial & Management Optimization, 2015, 11 (1) : 241-264. doi: 10.3934/jimo.2015.11.241

[10]

Zhimin Zhang. On a risk model with randomized dividend-decision times. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1041-1058. doi: 10.3934/jimo.2014.10.1041

[11]

Linyi Qian, Lyu Chen, Zhuo Jin, Rongming Wang. Optimal liability ratio and dividend payment strategies under catastrophic risk. Journal of Industrial & Management Optimization, 2018, 13 (5) : 1-19. doi: 10.3934/jimo.2018015

[12]

Badr-eddine Berrhazi, Mohamed El Fatini, Tomás Caraballo, Roger Pettersson. A stochastic SIRI epidemic model with Lévy noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2415-2431. doi: 10.3934/dcdsb.2018057

[13]

Linyi Qian, Wei Wang, Rongming Wang. Risk-minimizing portfolio selection for insurance payment processes under a Markov-modulated model. Journal of Industrial & Management Optimization, 2013, 9 (2) : 411-429. doi: 10.3934/jimo.2013.9.411

[14]

Dingjun Yao, Kun Fan. Optimal risk control and dividend strategies in the presence of two reinsurers: Variance premium principle. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1055-1083. doi: 10.3934/jimo.2017090

[15]

Xiangjun Wang, Jianghui Wen, Jianping Li, Jinqiao Duan. Impact of $\alpha$-stable Lévy noise on the Stommel model for the thermohaline circulation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1575-1584. doi: 10.3934/dcdsb.2012.17.1575

[16]

Rachel Chen, Jianqiang Hu, Yijie Peng. Simulation of Lévy-Driven models and its application in finance. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 749-765. doi: 10.3934/naco.2012.2.749

[17]

Xingchun Wang, Yongjin Wang. Hedging strategies for discretely monitored Asian options under Lévy processes. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1209-1224. doi: 10.3934/jimo.2014.10.1209

[18]

Chaman Kumar, Sotirios Sabanis. On tamed milstein schemes of SDEs driven by Lévy noise. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 421-463. doi: 10.3934/dcdsb.2017020

[19]

Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221

[20]

Kegui Chen, Xinyu Wang, Min Huang, Wai-Ki Ching. Compensation plan, pricing and production decisions with inventory-dependent salvage value, and asymmetric risk-averse sales agent. Journal of Industrial & Management Optimization, 2018, 13 (5) : 1-25. doi: 10.3934/jimo.2018013

2017 Impact Factor: 0.994

Metrics

  • PDF downloads (68)
  • HTML views (357)
  • Cited by (0)

Other articles
by authors

[Back to Top]