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Switching game of backward stochastic differential equations and associated system of obliquely reflected backward stochastic differential equations
November  2015, 35(11): 5467-5498. doi: 10.3934/dcds.2015.35.5467

A Dynkin game under Knightian uncertainty

 1 Graduate Department of Financial Engineering, Ajou University, Suwon 443-749, South Korea 2 School of Mathematical Science, Fudan University, Shanghai 200433 3 School of Mathematical Science, South China Normal University, Guangzhou 510631, China

Received  November 2012 Revised  October 2014 Published  May 2015

We study a zero-sum Dynkin game under Knghtian uncertainty. The associated Hamiton-Jacobi-Bellman-Isaacs equation takes the form of a semi-linear backward stochastic partial differential variational inequality (SBSPDVI). We establish existence and uniqueness of a strong solution by using the Banach fixed point theorem and a comparison theorem. A solution to the SBSPDVI is used to construct a saddle point of the Dynkin game. In order to establish this verification we use the generalized Itó-Kunita-Wentzell formula developed by Yang and Tang (2013).
Citation: Hyeng Keun Koo, Shanjian Tang, Zhou Yang. A Dynkin game under Knightian uncertainty. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5467-5498. doi: 10.3934/dcds.2015.35.5467
References:
 [1] A. Bensoussan and A. Friedman, Nonlinear variational inequalities and differential games with stopping times, Journal of Functional Analysis, 16 (1974), 305-352. doi: 10.1016/0022-1236(74)90076-7.  Google Scholar [2] J. Bismut, Sur un problème de Dynkin, Z. Warsch. V. Geb., 39 (1977), 31-53. doi: 10.1007/BF01844871.  Google Scholar [3] Z. Chen and L. Epstein, Ambiguity, risk and asset return in continuous time, Econometrica, 70 (2002), 1403-1443. doi: 10.1111/1468-0262.00337.  Google Scholar [4] Z. Chen, W. Tian and G. Zhao, Optimal stopping rule under ambiguity in continuous time,, submitted., ().   Google Scholar [5] Z. Chen and F. Riedel, Optimal stopping under ambiguity in continuous time, Mathem. Finan. Econom., 7 (2013), 29-68. doi: 10.1007/s11579-012-0081-6.  Google Scholar [6] K. Choi and G. Shim, Disutility, optimal retirement, and portfolio selection, Mathematical Finance, 16 (2006), 443-467. doi: 10.1111/j.1467-9965.2006.00278.x.  Google Scholar [7] J. Cvitanié and I. Karatzas, Backward stochastic differential games with reflections and Dynkin games, SIAM J. Control Optim., 24 (1996), 2024-2056. doi: 10.1214/aop/1041903216.  Google Scholar [8] F. Delbaen, S. Peng and E. Rosazza Gianin, Representation of the penalty term of dynamic concave utilities, Finance and Stochastics, 14 (2010), 449-472. doi: 10.1007/s00780-009-0119-7.  Google Scholar [9] A. Dixit and R. Pindyck, Investment Under Uncertainty, Princeton University Press, New Jersey, 1994. Google Scholar [10] K. Du and S. Tang, Strong solution of backward stochastic partial differential equations in $C^2$ domains, Probab. Theory Relat. Fields, 154 (2012), 255-285. doi: 10.1007/s00440-011-0369-0.  Google Scholar [11] E. Dynkin, Game variant of a problem on optimal stopping, Soviet Mathematics Doklady, 10 (1967), 270-274. Google Scholar [12] E. Dynkin and A. Yushkevich, Theorems and Problems in Markov Processes, Prenum press, New York, 1968. Google Scholar [13] D. Ellsberg, Risk, ambiguity, and Savage axioms, Quart. J. of Econom., 75 (1961), 643-669. Google Scholar [14] E. Fahri and S. Panages, Saving and investing for early retirement: A theoretical analysis, J. Finan. Econom., 83 (2007), 87-121. Google Scholar [15] I. Gilboa and D. Schmeidler, Maximin expected utility with non-unique prior, J. Math. Econom., 18 (1989), 141-153. doi: 10.1016/0304-4068(89)90018-9.  Google Scholar [16] S. Hamadène and J. Zhang, The continuous-time non-zero sum Dynkin game problem and application in game options, SIAM J. Control Optim., 48 (2010), 3659-3669. doi: 10.1137/080738933.  Google Scholar [17] L. Hansen and T. Sargen, Chapter 20-Wanting robustness in macroeconomics, in Handbook of Monetary Economics, Vol. 3 (eds. B. M. Friedman and M. Woodford), IOS Press, 2010, 1097-1157. doi: 10.1016/B978-0-444-53454-5.00008-6.  Google Scholar [18] I. Karatzas and S. E. Shreve, Methods of Mathematical Finance, Springer-Verlag, New York, 1998. doi: 10.1007/b98840.  Google Scholar [19] I. Karatzas and H. Wang, Connections between bounded variation control and Dynkin games, in Optimal Control and Partial Differential Equations, (eds. J. Menaldi, E. Rofman and A. Sulem), IOS Press, 2001, 363-373. Google Scholar [20] Karlin and Taylor, Second Course of Stochastic Processes, Jonn, Wiley & Sons, New York, 1985. Google Scholar [21] F. Knight, Risk, Uncertainty, and Profit, Houghton Mifflin, New York, 1921. doi: 10.1017/CBO9780511817410.005.  Google Scholar [22] F. Maccheroni, M. Marinacci and A. Rustichini, Ambiguity aversion, robustness, and the variational representation of preferences, Econometrica, 74 (2006), 1447-1498. doi: 10.1111/j.1468-0262.2006.00716.x.  Google Scholar [23] J. Qiu and S. Tang, On backward doubly stochastic differential evolutionary system,, preprint, ().   Google Scholar [24] F. Riedel, Optimal stopping with multiple priors, Econometrica, 77 (2009), 857-908. doi: 10.3982/ECTA7594.  Google Scholar [25] S. Tang and H. Koo, Options: A Framework of Optimal Switching, in Real, Ch. 2 of New Frontiers of Financial Engineering, (eds. H. Koo), IOS Press, 2011, 17-27.  Google Scholar [26] N. Touzi and N. Vieille, Continuous-time Dynkin game with mixed strategies, SIAM J. Control Optim., 41 (2002), 1073-1088. doi: 10.1137/S0363012900369812.  Google Scholar [27] Z. Yang and S. Tang, Dynkin game of stochastic differential equations with random coefficients, and associated backward stochastic partial differential variational inequality, SIAM J. Control Optim., 51 (2013), 64-95. doi: 10.1137/110850980.  Google Scholar

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References:
 [1] A. Bensoussan and A. Friedman, Nonlinear variational inequalities and differential games with stopping times, Journal of Functional Analysis, 16 (1974), 305-352. doi: 10.1016/0022-1236(74)90076-7.  Google Scholar [2] J. Bismut, Sur un problème de Dynkin, Z. Warsch. V. Geb., 39 (1977), 31-53. doi: 10.1007/BF01844871.  Google Scholar [3] Z. Chen and L. Epstein, Ambiguity, risk and asset return in continuous time, Econometrica, 70 (2002), 1403-1443. doi: 10.1111/1468-0262.00337.  Google Scholar [4] Z. Chen, W. Tian and G. Zhao, Optimal stopping rule under ambiguity in continuous time,, submitted., ().   Google Scholar [5] Z. Chen and F. Riedel, Optimal stopping under ambiguity in continuous time, Mathem. Finan. Econom., 7 (2013), 29-68. doi: 10.1007/s11579-012-0081-6.  Google Scholar [6] K. Choi and G. Shim, Disutility, optimal retirement, and portfolio selection, Mathematical Finance, 16 (2006), 443-467. doi: 10.1111/j.1467-9965.2006.00278.x.  Google Scholar [7] J. Cvitanié and I. Karatzas, Backward stochastic differential games with reflections and Dynkin games, SIAM J. Control Optim., 24 (1996), 2024-2056. doi: 10.1214/aop/1041903216.  Google Scholar [8] F. Delbaen, S. Peng and E. Rosazza Gianin, Representation of the penalty term of dynamic concave utilities, Finance and Stochastics, 14 (2010), 449-472. doi: 10.1007/s00780-009-0119-7.  Google Scholar [9] A. Dixit and R. Pindyck, Investment Under Uncertainty, Princeton University Press, New Jersey, 1994. Google Scholar [10] K. Du and S. Tang, Strong solution of backward stochastic partial differential equations in $C^2$ domains, Probab. Theory Relat. Fields, 154 (2012), 255-285. doi: 10.1007/s00440-011-0369-0.  Google Scholar [11] E. Dynkin, Game variant of a problem on optimal stopping, Soviet Mathematics Doklady, 10 (1967), 270-274. Google Scholar [12] E. Dynkin and A. Yushkevich, Theorems and Problems in Markov Processes, Prenum press, New York, 1968. Google Scholar [13] D. Ellsberg, Risk, ambiguity, and Savage axioms, Quart. J. of Econom., 75 (1961), 643-669. Google Scholar [14] E. Fahri and S. Panages, Saving and investing for early retirement: A theoretical analysis, J. Finan. Econom., 83 (2007), 87-121. Google Scholar [15] I. Gilboa and D. Schmeidler, Maximin expected utility with non-unique prior, J. Math. Econom., 18 (1989), 141-153. doi: 10.1016/0304-4068(89)90018-9.  Google Scholar [16] S. Hamadène and J. Zhang, The continuous-time non-zero sum Dynkin game problem and application in game options, SIAM J. Control Optim., 48 (2010), 3659-3669. doi: 10.1137/080738933.  Google Scholar [17] L. Hansen and T. Sargen, Chapter 20-Wanting robustness in macroeconomics, in Handbook of Monetary Economics, Vol. 3 (eds. B. M. Friedman and M. Woodford), IOS Press, 2010, 1097-1157. doi: 10.1016/B978-0-444-53454-5.00008-6.  Google Scholar [18] I. Karatzas and S. E. Shreve, Methods of Mathematical Finance, Springer-Verlag, New York, 1998. doi: 10.1007/b98840.  Google Scholar [19] I. Karatzas and H. Wang, Connections between bounded variation control and Dynkin games, in Optimal Control and Partial Differential Equations, (eds. J. Menaldi, E. Rofman and A. Sulem), IOS Press, 2001, 363-373. Google Scholar [20] Karlin and Taylor, Second Course of Stochastic Processes, Jonn, Wiley & Sons, New York, 1985. Google Scholar [21] F. Knight, Risk, Uncertainty, and Profit, Houghton Mifflin, New York, 1921. doi: 10.1017/CBO9780511817410.005.  Google Scholar [22] F. Maccheroni, M. Marinacci and A. Rustichini, Ambiguity aversion, robustness, and the variational representation of preferences, Econometrica, 74 (2006), 1447-1498. doi: 10.1111/j.1468-0262.2006.00716.x.  Google Scholar [23] J. Qiu and S. Tang, On backward doubly stochastic differential evolutionary system,, preprint, ().   Google Scholar [24] F. Riedel, Optimal stopping with multiple priors, Econometrica, 77 (2009), 857-908. doi: 10.3982/ECTA7594.  Google Scholar [25] S. Tang and H. Koo, Options: A Framework of Optimal Switching, in Real, Ch. 2 of New Frontiers of Financial Engineering, (eds. H. Koo), IOS Press, 2011, 17-27.  Google Scholar [26] N. Touzi and N. Vieille, Continuous-time Dynkin game with mixed strategies, SIAM J. Control Optim., 41 (2002), 1073-1088. doi: 10.1137/S0363012900369812.  Google Scholar [27] Z. Yang and S. Tang, Dynkin game of stochastic differential equations with random coefficients, and associated backward stochastic partial differential variational inequality, SIAM J. Control Optim., 51 (2013), 64-95. doi: 10.1137/110850980.  Google Scholar
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