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Fully nonlinear stochastic and rough PDEs: Classical and viscosity solutions
Efficient hedging under ambiguity in continuous time
Princeton University, Department of Operations Research and Financial Engineering, Princeton 08540, NJ, USA |
References:
[1] |
Acciaio, B., Beiglböck, M., Penkner, F., Schachermayer, W.: A model-free version of the fundamental theorem of asset pricing and the super-replication theorem. Math. Finance. 26(2), 233–251 (2016), |
[2] |
Aliprantis, C.D., Border, K.C.:: Infinite Dimensional Analysis: a Hitchhiker’s Guide, 3rd ed. Springer(2006), |
[3] |
Arai, T.: Convex risk measures on orlicz spaces: inf-convolution and shortfall. Math. Finan. Econ. 3, 73– 88 (2010), |
[4] |
Bartl, D., Drapeau, S., Tangpi, L.: Computational aspects of robust optimized certainty equivalents and option pricing. Math. Finance. 30(1), 287–09 (2020), |
[5] |
Bartl, D., Kupper, M., Prömel, D.J., Tangpi, L.: Duality for pathwise superhedging in continuous time. Finance Stoch. 23(3), 697–728 (2019), |
[6] |
Bartl, D., Kupper, M., Neufeld, A.: Pathwise superhedging on prediction sets. Finance Stoch. 24, 215–48(2020), |
[7] |
Becherer, D., Kentia, K.: Good deal hedging and valuation under combined uncertainty about drift and volatility. Probab. Uncertain. Quant. Risk. 2(13) (2017), |
[8] |
Beiglböck, M., Henry-Labordère, P., Penkner, F.: Model-independent bounds for option prices – a mass transport approach. Finance Stoch. 17(3), 477–501 (2013), |
[9] |
Ben-Tal, A., Taboulle, M.: An old-new concept of convex risk measures: The optimized certainty equivalent. Math. Finance. 17, 449–476 (2007), |
[10] |
Bion-Nadal, J., Di Nunno, G.: Dynamic no-good-deal pricing measures and extension theorems for linear operators on L∞. Finance Stoch. 17(3), 587–613 (2013), |
[11] |
Bion-Nadal, J., Kervarec, M.: Risk Measuring under Model Uncertainty. Ann. Appl. Probab. 22(1), 213– 238 (2012), |
[12] |
Burzoni, M.: Arbitrage and hedging in model independent markets with frictions. SIAM J. Financial Math. 7(1), 812–844 (2016), |
[13] |
Burzoni, M., Frittelli, M., Maggis, M.: Model-free superhedging duality. Ann. Appl. Probab. 27(3), 1452– 1477 (2017), |
[14] |
Cheridito, P., Kupper, M., Tangpi, L.: Representation of increasing convex functionals with countably additive measures. Preprint (2015), |
[15] |
Cheridito, P., Kupper, M., Tangpi, L.: Duality formulas for robust pricing and hedging in discrete time.SIAM J. Financial Math. 8(1), 738–765 (2017), |
[16] |
Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math.Ann. 300(3), 463–520 (1994), |
[17] |
Dellacherie, C., Meyer, P.-A.:: Probabilities and Potential. B. North-Holland Mathematics Studies, vol. 72, p. 463. North-Holland Publishing Co., Amsterdam (1982). Theory of martingales, Translated from the French by J. P. Wilson Denis, L., Martini, C.: A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab. 16, 827–852 (2006), |
[18] |
Dolinsky, Y., Soner, H.M.: Martingale optimal transport and robust hedging in continuous time. Probab.Theory Relat. Fields. 160(1-2), 391–427 (2014), |
[19] |
Fan, K.: Minimax theorems. Proc. Nat. Acad. Sci. U.S.A. 39, 42–47 (1953), |
[20] |
Föllmer, H., Leukert, P.: Quantile hedging. Finance Stoch. 3(3), 251–273 (1999), |
[21] |
Föllmer, H., Leukert, P.: Efficient hedging: Cost versus shortfall risk. Finance Stoch. 4, 117–146 (2000), |
[22] |
Hou, Z., Obłój, J.: On robust pricing-hedging duality in continuous time. Finance Stoch. 22(3), 511–567(2018), |
[23] |
Jouini, E., Schachermayer, W., Touzi, N.: Law invariant risk measures have the fatou property. Adv. Math.Econ. 9, 49–71 (2006), |
[24] |
Kaina, M., Rüschendorf, L.: On convex risk measures on lp-spaces. Math. Meth. Oper. Res. 69(3), 475– 495 (2009), |
[25] |
Karandikar, R.L.: On quadratic variation process of a continuous martingale. Illinois J. Math. 27, 178–181(1983), |
[26] |
Karatzas, I., Shreve, S.E.:: Brownian Motion and Stochastic Calculus (Graduate Texts in Mathematics).Springer, New York (2004), |
[27] |
Kramkov, D., Schachermayer, W.: The Asymptotic Elasticity of Utility Functions and Optimal Investment in Incomplete Market. Ann. Appl. Probab. 9(3), 904–950 (1999), |
[28] |
Kupper, M., Schachermayer, W.: Representation results for law invariant time consistent functions. Math.Financ. Econ. 2(3), 189–210 (2009), |
[29] |
Neufeld, A., Nutz, M.: Superreplication under Volatility Uncertainty for Measurable Claims. Electron. J.Probab. 18(48), 1–14 (2013), |
[30] |
Peng, S.: Nonlinear expectations and stochastic calculus under uncertainty. arXiv Preprint 1002.4546(2010), |
[31] |
Rudloff, B.: Convex hedging in incomplete markets. Appl. Math. Finance. 14(5), 437–452 (2007), |
[32] |
Soner, H.M., Touzi, N., Zhang, J.: Martingale representation theorem for the G-expectation. Stoch. Proc.Appl. 121(2), 265–287 (2011a), |
[33] |
Soner, M.H., Touzi, N., Zhang, J.: Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16(67) (2011b), |
[34] |
Soner, H.M., Touzi, N., Zhang, J.: Dual formulation of second order target problems. Ann. Appl. Probab. 23(1), 308–347 (2013), |
[35] |
Tangpi, L.: Dual Representation of Convex Increasing Functionals with Applications to Finance. PhD thesis, University of Konstanz (2015), |
[36] |
Wisniewski, A.: The structure of measurable mappings on metric spaces. Proc. A.M.S. 122(1), 147–150(1994), |
show all references
References:
[1] |
Acciaio, B., Beiglböck, M., Penkner, F., Schachermayer, W.: A model-free version of the fundamental theorem of asset pricing and the super-replication theorem. Math. Finance. 26(2), 233–251 (2016), |
[2] |
Aliprantis, C.D., Border, K.C.:: Infinite Dimensional Analysis: a Hitchhiker’s Guide, 3rd ed. Springer(2006), |
[3] |
Arai, T.: Convex risk measures on orlicz spaces: inf-convolution and shortfall. Math. Finan. Econ. 3, 73– 88 (2010), |
[4] |
Bartl, D., Drapeau, S., Tangpi, L.: Computational aspects of robust optimized certainty equivalents and option pricing. Math. Finance. 30(1), 287–09 (2020), |
[5] |
Bartl, D., Kupper, M., Prömel, D.J., Tangpi, L.: Duality for pathwise superhedging in continuous time. Finance Stoch. 23(3), 697–728 (2019), |
[6] |
Bartl, D., Kupper, M., Neufeld, A.: Pathwise superhedging on prediction sets. Finance Stoch. 24, 215–48(2020), |
[7] |
Becherer, D., Kentia, K.: Good deal hedging and valuation under combined uncertainty about drift and volatility. Probab. Uncertain. Quant. Risk. 2(13) (2017), |
[8] |
Beiglböck, M., Henry-Labordère, P., Penkner, F.: Model-independent bounds for option prices – a mass transport approach. Finance Stoch. 17(3), 477–501 (2013), |
[9] |
Ben-Tal, A., Taboulle, M.: An old-new concept of convex risk measures: The optimized certainty equivalent. Math. Finance. 17, 449–476 (2007), |
[10] |
Bion-Nadal, J., Di Nunno, G.: Dynamic no-good-deal pricing measures and extension theorems for linear operators on L∞. Finance Stoch. 17(3), 587–613 (2013), |
[11] |
Bion-Nadal, J., Kervarec, M.: Risk Measuring under Model Uncertainty. Ann. Appl. Probab. 22(1), 213– 238 (2012), |
[12] |
Burzoni, M.: Arbitrage and hedging in model independent markets with frictions. SIAM J. Financial Math. 7(1), 812–844 (2016), |
[13] |
Burzoni, M., Frittelli, M., Maggis, M.: Model-free superhedging duality. Ann. Appl. Probab. 27(3), 1452– 1477 (2017), |
[14] |
Cheridito, P., Kupper, M., Tangpi, L.: Representation of increasing convex functionals with countably additive measures. Preprint (2015), |
[15] |
Cheridito, P., Kupper, M., Tangpi, L.: Duality formulas for robust pricing and hedging in discrete time.SIAM J. Financial Math. 8(1), 738–765 (2017), |
[16] |
Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math.Ann. 300(3), 463–520 (1994), |
[17] |
Dellacherie, C., Meyer, P.-A.:: Probabilities and Potential. B. North-Holland Mathematics Studies, vol. 72, p. 463. North-Holland Publishing Co., Amsterdam (1982). Theory of martingales, Translated from the French by J. P. Wilson Denis, L., Martini, C.: A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab. 16, 827–852 (2006), |
[18] |
Dolinsky, Y., Soner, H.M.: Martingale optimal transport and robust hedging in continuous time. Probab.Theory Relat. Fields. 160(1-2), 391–427 (2014), |
[19] |
Fan, K.: Minimax theorems. Proc. Nat. Acad. Sci. U.S.A. 39, 42–47 (1953), |
[20] |
Föllmer, H., Leukert, P.: Quantile hedging. Finance Stoch. 3(3), 251–273 (1999), |
[21] |
Föllmer, H., Leukert, P.: Efficient hedging: Cost versus shortfall risk. Finance Stoch. 4, 117–146 (2000), |
[22] |
Hou, Z., Obłój, J.: On robust pricing-hedging duality in continuous time. Finance Stoch. 22(3), 511–567(2018), |
[23] |
Jouini, E., Schachermayer, W., Touzi, N.: Law invariant risk measures have the fatou property. Adv. Math.Econ. 9, 49–71 (2006), |
[24] |
Kaina, M., Rüschendorf, L.: On convex risk measures on lp-spaces. Math. Meth. Oper. Res. 69(3), 475– 495 (2009), |
[25] |
Karandikar, R.L.: On quadratic variation process of a continuous martingale. Illinois J. Math. 27, 178–181(1983), |
[26] |
Karatzas, I., Shreve, S.E.:: Brownian Motion and Stochastic Calculus (Graduate Texts in Mathematics).Springer, New York (2004), |
[27] |
Kramkov, D., Schachermayer, W.: The Asymptotic Elasticity of Utility Functions and Optimal Investment in Incomplete Market. Ann. Appl. Probab. 9(3), 904–950 (1999), |
[28] |
Kupper, M., Schachermayer, W.: Representation results for law invariant time consistent functions. Math.Financ. Econ. 2(3), 189–210 (2009), |
[29] |
Neufeld, A., Nutz, M.: Superreplication under Volatility Uncertainty for Measurable Claims. Electron. J.Probab. 18(48), 1–14 (2013), |
[30] |
Peng, S.: Nonlinear expectations and stochastic calculus under uncertainty. arXiv Preprint 1002.4546(2010), |
[31] |
Rudloff, B.: Convex hedging in incomplete markets. Appl. Math. Finance. 14(5), 437–452 (2007), |
[32] |
Soner, H.M., Touzi, N., Zhang, J.: Martingale representation theorem for the G-expectation. Stoch. Proc.Appl. 121(2), 265–287 (2011a), |
[33] |
Soner, M.H., Touzi, N., Zhang, J.: Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16(67) (2011b), |
[34] |
Soner, H.M., Touzi, N., Zhang, J.: Dual formulation of second order target problems. Ann. Appl. Probab. 23(1), 308–347 (2013), |
[35] |
Tangpi, L.: Dual Representation of Convex Increasing Functionals with Applications to Finance. PhD thesis, University of Konstanz (2015), |
[36] |
Wisniewski, A.: The structure of measurable mappings on metric spaces. Proc. A.M.S. 122(1), 147–150(1994), |
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