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A general optimal multiple stopping problem with an application to swing options. (English) Zbl 1327.60098
Summary: In their paper [Math. Finance 18, No. 2, 239–268 (2008; Zbl 1133.91499)], R. Carmona and N. Touzi studied an optimal multiple stopping time problem in a market where the price process is continuous. In this article, we generalize their results when the price process is allowed to jump. Also, we generalize the problem associated to the valuation of swing options to the context of jump diffusion processes. We relate our problem to a sequence of ordinary stopping time problems. We characterize the value function of each ordinary stopping time problem as the unique viscosity solution of the associated Hamilton-Jacobi-Bellman variational inequality.

MSC:
60G40 Stopping times; optimal stopping problems; gambling theory
60J60 Diffusion processes
60J75 Jump processes (MSC2010)
91G80 Financial applications of other theories
91B70 Stochastic models in economics
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
49J40 Variational inequalities
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References:
[1] DOI: 10.1137/09077076X · Zbl 1270.91090
[2] DOI: 10.1007/s00780-010-0134-8 · Zbl 1303.91167
[3] DOI: 10.1239/aap/1158684999 · Zbl 1114.60033
[4] Bender C., Math. Finance (2015)
[5] DOI: 10.1142/S0219024906004037 · Zbl 1139.60328
[6] DOI: 10.1287/moor.1070.0301 · Zbl 1221.60061
[7] DOI: 10.1111/j.1467-9965.2007.00331.x · Zbl 1133.91499
[8] Crandall M.G., Diff. Int. Eqn. 3 pp 1001– (1990)
[9] DOI: 10.1090/S0002-9947-1983-0690039-8
[10] DOI: 10.1142/S0219024905002895 · Zbl 1100.91042
[11] Dellacherie C., Probabilités et potentiel. Chapitres V à VIII : théorie des martingales (1980)
[12] El Karoui, N. 1981. Les aspects probabilistes du controle stochastique. Lecture Notes in Mathematics, 876. New York: Springer-Verlag, 73–238. · Zbl 0472.60002
[13] Gimbert F., Ricerche Mat. 33 pp 315– (1984)
[14] DOI: 10.1007/b98840
[15] DOI: 10.1214/10-AAP727 · Zbl 1235.60040
[16] DOI: 10.1080/03605308308820297 · Zbl 0716.49022
[17] Maingueneau M.A., Séminaire de probabilités (Strasbourg) 12 pp 457– (1978)
[18] DOI: 10.1007/978-3-540-69826-5
[19] Peskir G., Optimal Stopping and Free-Boundary Problems (2006) · Zbl 1115.60001
[20] DOI: 10.1007/BF02683325 · Zbl 0866.60038
[21] Pham H., Journal of Mathematical Systems, Estimation, and Control 8 (1) pp 1– (1998)
[22] Protter P.E., Stochastic Integration and Differential Equations (2004) · Zbl 1041.60005
[23] DOI: 10.1017/CBO9781107590120
[24] Sayah A., Comm. P.D.E. 10 pp 1057– (1991)
[25] Soner H.M., SIAM J. Control Optim. 24 pp 110– (1986)
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