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December  2015, 5(4): 807-826. doi: 10.3934/mcrf.2015.5.807

## Generalization on optimal multiple stopping with application to swing options with random exercise rights number

 1 Département de Mathématiques, Institut Supérieur d'Informatique et de Mathématiques de Monastir, Avenue de la Korniche, B.P. 223, 5000 Monastir, Tunisia, Tunisia

Received  December 2014 Revised  April 2015 Published  October 2015

This paper develops the theory of optimal multiple stopping times expected value problems by stating, proving, and applying a dynamic programming principle for the case in which both the reward process and the number of stopping times are stochastic. This case comes up in practice when valuing swing options, which are somewhat common in commodity trading. We believe our results significantly advance the study of option pricing.
Citation: Noureddine Jilani Ben Naouara, Faouzi Trabelsi. Generalization on optimal multiple stopping with application to swing options with random exercise rights number. Mathematical Control and Related Fields, 2015, 5 (4) : 807-826. doi: 10.3934/mcrf.2015.5.807
##### References:
 [1] C. Blanchet-Scalliet, N. El-Karoui, M. Jeanblanc and L. Martellini, Optimal investment and consumption decisions when time-horizon is uncertain, Journal of Mathematical Economics, 44 (2008), 1100-1113. doi: 10.1016/j.jmateco.2007.09.004. [2] R. Carmona and S. Dayanik, Optimal multiple stopping of linear diffusions, Mathematics of Operations Research, 33 (2008), 446-460. doi: 10.1287/moor.1070.0301. [3] R. Carmona and N. Touzi, Optimal multiple stopping and valuation of swing options, Mathematical Finance, 18 (2008), 239-268. doi: 10.1111/j.1467-9965.2007.00331.x. [4] N. Chaidee and K. Neammanee, Berry-Esseen bound for independent random sum via Stein's method, International Mathematical Forum, 3 (2008), 721-738. [5] N. Chaidee and M. Tuntapthai, Berry-Esseen bounds for random sums of non-i.i.d. random variables, International Mathematical Forum, 4 (2009), 1281-1288. [6] S. Christensen, A. Irle and S. Jürgens, Optimal multiple stopping with random waiting times, Sequential Analysis: Design Methods and Applications, 32 (2013), 297-318. doi: 10.1080/07474946.2013.803814. [7] S. Dayanik and I. Karatzas, On the optimal stopping times problem for one-dimensional diffusions, Stochastic Processes and their Applications, 9 (2003), 342-351. [8] E. B. Dynkin, Markov Processes: Theorems and Problems, 1st edition, Plenum Press, New York, 1969. [9] R. Elliott, M. Jeanblanc and M. Yor, On models of default risk, Mathematical Finance, 10 (2000), 179-195. doi: 10.1111/1467-9965.00088. [10] S. W. He, J. G. Wang and J. A. Yan, Semimartingale Theory and Stochastic Calculus, Science Press, Beijing, CRC Press, Boca Raton, 1992. [11] K. Itô and H. P. McKean, Diffusion Processes and Their Sample Paths, 1st edition, Springer-Verlag, Berlin, 1974. [12] N. Jilani Ben Naouara and F. Trabelsi, Biological application of optimal stopping, Int. J. of Mathematical Modelling and Numerical Optimisation, 5 (2014), 229-264. [13] N. Jilani Ben Naouara and F. Trabelsi, General undiscounted non-linear optimal multiple stopping of linear diffusions with boundary classification, to appear in Int. J. of Mathematics in Operational Research. [14] S. Karlin and H. Taylor, A Second Course in Stochastic Processes, Academic press, San Diego, 1981. [15] M. Kobylanski, M. C. Quenez and E. Rouy, Optimal multiple stopping time problem, The Annals of Applied Probability, 21 (2011), 1365-1399. doi: 10.1214/10-AAP727. [16] M. Pointier, Pricing Rules Under Asymmetric Information, http://www.math.univ-toulouse.fr/ pontier/squfirenze.pdf. [17] M. Tomomi and A. Katsunori, Lower bounds for Bruss' odds problem with multiple stopping, preprint, arXiv:1204.5537. [18] F. Trabelsi, Study of undiscounted non-linear optimal multiple stopping times problems on unbounded intervals, Int. J. Operational Research, 5 (2013), 225-254. doi: 10.1504/IJMOR.2013.052462. [19] F. Trabelsi and M. B. Zoghlami, On undiscounted non-linear optimal multiple stopping, Int. J. Operational Research, 14 (2012), 387-416. doi: 10.1504/IJOR.2012.047512. [20] A. B. Zeghal and M. Mnif, Optimal multiple stopping and valuation of swing options in Lévy models, Int. J. Theoretical and Applied Finance, 9 (2006), 1267-1297. doi: 10.1142/S0219024906004037.

show all references

##### References:
 [1] C. Blanchet-Scalliet, N. El-Karoui, M. Jeanblanc and L. Martellini, Optimal investment and consumption decisions when time-horizon is uncertain, Journal of Mathematical Economics, 44 (2008), 1100-1113. doi: 10.1016/j.jmateco.2007.09.004. [2] R. Carmona and S. Dayanik, Optimal multiple stopping of linear diffusions, Mathematics of Operations Research, 33 (2008), 446-460. doi: 10.1287/moor.1070.0301. [3] R. Carmona and N. Touzi, Optimal multiple stopping and valuation of swing options, Mathematical Finance, 18 (2008), 239-268. doi: 10.1111/j.1467-9965.2007.00331.x. [4] N. Chaidee and K. Neammanee, Berry-Esseen bound for independent random sum via Stein's method, International Mathematical Forum, 3 (2008), 721-738. [5] N. Chaidee and M. Tuntapthai, Berry-Esseen bounds for random sums of non-i.i.d. random variables, International Mathematical Forum, 4 (2009), 1281-1288. [6] S. Christensen, A. Irle and S. Jürgens, Optimal multiple stopping with random waiting times, Sequential Analysis: Design Methods and Applications, 32 (2013), 297-318. doi: 10.1080/07474946.2013.803814. [7] S. Dayanik and I. Karatzas, On the optimal stopping times problem for one-dimensional diffusions, Stochastic Processes and their Applications, 9 (2003), 342-351. [8] E. B. Dynkin, Markov Processes: Theorems and Problems, 1st edition, Plenum Press, New York, 1969. [9] R. Elliott, M. Jeanblanc and M. Yor, On models of default risk, Mathematical Finance, 10 (2000), 179-195. doi: 10.1111/1467-9965.00088. [10] S. W. He, J. G. Wang and J. A. Yan, Semimartingale Theory and Stochastic Calculus, Science Press, Beijing, CRC Press, Boca Raton, 1992. [11] K. Itô and H. P. McKean, Diffusion Processes and Their Sample Paths, 1st edition, Springer-Verlag, Berlin, 1974. [12] N. Jilani Ben Naouara and F. Trabelsi, Biological application of optimal stopping, Int. J. of Mathematical Modelling and Numerical Optimisation, 5 (2014), 229-264. [13] N. Jilani Ben Naouara and F. Trabelsi, General undiscounted non-linear optimal multiple stopping of linear diffusions with boundary classification, to appear in Int. J. of Mathematics in Operational Research. [14] S. Karlin and H. Taylor, A Second Course in Stochastic Processes, Academic press, San Diego, 1981. [15] M. Kobylanski, M. C. Quenez and E. Rouy, Optimal multiple stopping time problem, The Annals of Applied Probability, 21 (2011), 1365-1399. doi: 10.1214/10-AAP727. [16] M. Pointier, Pricing Rules Under Asymmetric Information, http://www.math.univ-toulouse.fr/ pontier/squfirenze.pdf. [17] M. Tomomi and A. Katsunori, Lower bounds for Bruss' odds problem with multiple stopping, preprint, arXiv:1204.5537. [18] F. Trabelsi, Study of undiscounted non-linear optimal multiple stopping times problems on unbounded intervals, Int. J. Operational Research, 5 (2013), 225-254. doi: 10.1504/IJMOR.2013.052462. [19] F. Trabelsi and M. B. Zoghlami, On undiscounted non-linear optimal multiple stopping, Int. J. Operational Research, 14 (2012), 387-416. doi: 10.1504/IJOR.2012.047512. [20] A. B. Zeghal and M. Mnif, Optimal multiple stopping and valuation of swing options in Lévy models, Int. J. Theoretical and Applied Finance, 9 (2006), 1267-1297. doi: 10.1142/S0219024906004037.
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