×

Root’s barrier, viscosity solutions of obstacle problems and reflected FBSDEs. (English) Zbl 1335.60059

Summary: We revisit works of H. Rost [Lect. Notes Math. 511, 194–208 (1976; Zbl 0339.60042)], B. Dupire [“Arbitrage bounds for volatility derivatives as free boundary problem”, presented at KTH, Stockholm (2005)] and A. M. G. Cox and J. Wang [“Optimal robust bounds for variance option”, Preprint, arXiv:1308.436] on connections between Root’s solution of the Skorokhod embedding problem and obstacle problems. We develop an approach based on viscosity sub- and supersolutions and an accompanying comparison principle. This gives new, constructive and simple proofs of the existence and minimality properties of Root type solutions as well as their complete characterization. The approach is self-contained and covers martingale diffusions with degenerate elliptic or time-dependent volatility as well as Rost’s reversed Root barriers; it also provides insights about the dynamics of general Skorokhod embeddings by identifying them as supersolutions of certain nonlinear PDEs.

MSC:

60G40 Stopping times; optimal stopping problems; gambling theory
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
35D40 Viscosity solutions to PDEs

Citations:

Zbl 0339.60042
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ankirchner, Stefan; Heyne, Gregor; Imkeller, Peter, A BSDE approach to the Skorokhod embedding problem for the Brownian motion with drift, Stoch. Dyn., 8, 1, 35-46 (2008) · Zbl 1148.60026
[3] Ankirchner, Stefan; Strack, Philipp, Skorokhod embeddings in bounded time, Stoch. Dyn., 11, 02-03, 215-226 (2011) · Zbl 1241.60021
[4] Barles, Guy, A new stability result for viscosity solutions of nonlinear parabolic equations with weak convergence in time, C. R. Math. Acad. Sci. Paris, 343, 3, 173-178 (2006) · Zbl 1102.35014
[5] Barles, G.; Daher, Ch.; Romano, M., Convergence of numerical schemes for parabolic equations arising in finance theory, Math. Models Methods Appl. Sci., 5, 1, 125-143 (1995) · Zbl 0822.65056
[7] Barles, G.; Souganidis, P. E., Convergence of approximation schemes for fully nonlinear second order equations, Asymptot. Anal., 4, 3, 271-283 (1991) · Zbl 0729.65077
[10] Bensoussan, A.; Lions, J. L., Applications of Variational Inequalities in Stochastic Control. Volume 12 (1982), North Holland
[11] Buckdahn, R.; Huang, J.; Li, J., Regularity properties for general HJB equations: a backward stochastic differential equation method, SIAM J. Control Optim., 50, 3, 1466-1501 (2012) · Zbl 1246.93123
[12] Chacon, R. V., Potential processes, Trans. Amer. Math. Soc., 226, 39-58 (1977) · Zbl 0366.60106
[13] Chacon, Rene, Barrier stopping times and the filling scheme (1985), University of Washington, (Ph.D. dissertation)
[14] Chacon, R.; Walsh, J., One-dimensional potential embedding, (Séminaire de Probabilités X Université de Strasbourg (1976)), 19-23 · Zbl 0329.60041
[17] Cox, Alexander M. G.; Wang, Jiajie, Root’s barrier: Construction, optimality and applications to variance options, Ann. Appl. Probab., 23, 3, 859-894 (2013) · Zbl 1266.91101
[18] Crandall, M. G.; Ishii, H.; Lions, P.-L., User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (NS), 27, 1, 1-67 (1992) · Zbl 0755.35015
[20] Dinges, Hermann, Stopping sequences, (Séminaire de Probabilités VIII Université de Strasbourg (1974), Springer), 27-36
[22] El Karoui, N.; Kapoudjian, C.; Pardoux, E.; Peng, S.; Quenez, M. C., Reflected solutions of backward SDEs, and related obstacle problems for PDEs, Ann. Probab., 25, 2, 702-737 (1997) · Zbl 0899.60047
[23] El Karoui, N.; Peng, S.; Quenez, M. C., Backward stochastic differential equations in finance, Math. Finance, 7, 1, 1-71 (1997) · Zbl 0884.90035
[24] Fleming, W. H.; Soner, H. M., (Controlled Markov Processes and Viscosity Solutions. Controlled Markov Processes and Viscosity Solutions, Stochastic Modelling and Applied Probability, vol. 25 (2006), Springer: Springer New York) · Zbl 1105.60005
[25] Galichon, A.; Henry-Labordère, P.; Touzi, N., A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options, Ann. Appl. Probab., 24, 1, 312-336 (2014), 02 · Zbl 1285.49012
[26] Gassiat, Paul; Mijatović, Aleksandar; Oberhauser, Harald, An integral equation for Root’s barrier and the generation of Brownian increments, Ann. Appl. Probab., 25, 4, 2039-2065 (2015), http://projecteuclid.org/euclid.aoap/1432212436 · Zbl 1328.60103
[28] Hirsch, F.; Roynette, B., A new proof of Kellerer’s theorem, ESAIM Probab. Stat., 16, 48-60 (2012) · Zbl 1277.60041
[29] Hobson, David G., Robust hedging of the lookback option, Finance Stoch., 2, 4, 329-347 (1998) · Zbl 0907.90023
[30] Hobson, D., The Skorokhod embedding problem and model-independent bounds for option prices, (Paris-Princeton Lectures on Mathematical Finance 2010. Paris-Princeton Lectures on Mathematical Finance 2010, Lecture Notes in Mathematics, vol. 2003 (2011), Springer: Springer Berlin, Heidelberg), 267-318 · Zbl 1214.91113
[31] Jakobsen, E. R., On the rate of convergence of approximation schemes for Bellman equations associated with optimal stopping time problems, Math. Models Methods Appl. Sci., 13, 05, 613-644 (2003) · Zbl 1050.35042
[32] Jakobsen, E. R.; Karlsen, K. H., Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate parabolic equations, J. Differential Equations, 183, 2, 497-525 (2002) · Zbl 1086.35061
[33] Kellerer, Hans G., Markov-komposition und eine Anwendung auf Martingale, Math. Ann., 198, 3, 99-122 (1972) · Zbl 0229.60049
[34] Kiefer, Jack, Skorohod embedding of multivariate rv’s, and the sample df, Z. Wahrscheinlichkeitstheor. Verwandte Geb., 24, 1, 1-35 (1972) · Zbl 0267.60034
[36] Loynes, R. M., Stopping times on Brownian motion: Some properties of Root’s construction, Z. Wahrscheinlichkeitstheor. Verwandte Geb., 16, 211-218 (1970) · Zbl 0193.45701
[37] McConnell, Terry R., The two-sided stefan problem with a spatially dependent latent heat, Trans. Amer. Math. Soc., 326, 2, 669-699 (1991) · Zbl 0754.35193
[38] Nualart, D., (The Malliavin Calculus and Related Topics. The Malliavin Calculus and Related Topics, Probability and its Applications (New York) (2006), Springer-Verlag: Springer-Verlag Berlin) · Zbl 1099.60003
[39] Nunziante, Diana, Existence and uniqueness of unbounded viscosity solutions of parabolic equations with discontinuous time-dependence, Nonlinear Anal., 18, 11, 1033-1062 (1992) · Zbl 0782.35037
[40] Obłój, J., The Skorokhod embedding problem and its offspring, Probab. Surv., 1, 321-390 (2004) · Zbl 1189.60088
[43] Root, D. H., The existence of certain stopping times on Brownian motion, Ann. Math. Statist., 40, 715-718 (1969) · Zbl 0174.21902
[44] Rost, H., Die Stoppverteilungen eines Markoff-Prozesses mit lokalendlichem Potential, Manuscripta Math., 3, 321-329 (1970) · Zbl 0205.45101
[45] Rost, H., The stopping distributions of a Markov Process, Invent. Math., 14, 1-16 (1971) · Zbl 0225.60025
[46] Rost, H., Skorokhod’s theorem for general Markov processes, (Transactions of the Sixth Prague Conference on Information Theory, Statistical Decision Functions, Random Processes (Tech. Univ. Prague, Prague, 1971; dedicated to the memory of Antonín Špaček) (1973), Academia: Academia Prague), 755-764
[47] Rost, H., Skorokhod stopping times of minimal variance, (Séminaire de Probabilités, X (Première Partie, Univ. Strasbourg, Strasbourg, Année Universitaire 1974/1975). Séminaire de Probabilités, X (Première Partie, Univ. Strasbourg, Strasbourg, Année Universitaire 1974/1975), Lecture Notes in Math., vol. 511 (1976), Springer: Springer Berlin), 194-208 · Zbl 0339.60042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.