Stochastic Target Games and Dynamic Programming via Regularized Viscosity Solutions

References

• Avellaneda M, Levy A, Parás A (1995) Pricing and hedging derivative securities in markets with uncertain volatilities. Appl. Math. Finance 2(2):73–88.Crossref, Google Scholar
• Barles G, Jakobsen ER (2002) On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations. M2AN Math. Model. Numer. Anal. 36(1):33–54.Crossref, Google Scholar
• Barles G, Souganidis PE (1991) Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Anal. 4(3):271–283.Crossref, Google Scholar
• Bayraktar E, Li J (2015) Stochastic Perron for stochastic target games. Ann. Appl. Probab. Forthcoming.Google Scholar
• Bouchard B, Nutz M (2015) Arbitrage and duality in nondominated discrete-time models. Ann. Appl. Probab. 25(2):823–859.Crossref, Google Scholar
• Bouchard B, Elie R, Touzi N (2009) Stochastic target problems with controlled loss. SIAM J. Control Optim. 48(5):3123–3150.Crossref, Google Scholar
• Bouchard B, Moreau L, Nutz M (2014) Stochastic target games with controlled loss. Ann. Appl. Probab. 24(3):899–934.Crossref, Google Scholar
• Buckdahn R, Li J (2008) Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations. SIAM J. Control Optim. 47(1):444–475.Crossref, Google Scholar
• Crandall MG, Ishii H, Lions PL (1992) User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27(1):1–67.Crossref, Google Scholar
• Dellacherie C, Meyer PA (1982) Probabilities and Potential B (North-Holland, Amsterdam).Google Scholar
• Denis L, Martini C (2006) A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab. 16(2):827–852.Crossref, Google Scholar
• Fleming WH, Souganidis PE (1989) On the existence of value functions of two-player, zero-sum stochastic differential games. Indiana Univ. Math. J. 38(2):293–314.Crossref, Google Scholar
• Ishii H (1995) On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions. Funkciala Ekvacioj 38(1):101–120.Google Scholar
• Jensen R (1988) The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations. Arch. Rational Mech. Anal. 101(1):1–27.Crossref, Google Scholar
• Krylov NV (2000) On the rate of convergence of finite-difference approximations for Bellman’s equations with variable coefficients. Probab. Theory Related Fields 117(1):1–16.Crossref, Google Scholar
• Neufeld A, Nutz M (2013) Superreplication under volatility uncertainty for measurable claims. Electron. J. Probab. 18(48):1–14.Google Scholar
• Neveu J (1975) Discrete-Parameter Martingales (North-Holland, Amsterdam).Google Scholar
• Pardoux E (1998) Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order. Progr. Probab. 42:79–128.Google Scholar
• Peng S (1992) A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation. Stochastics Stochastics Rep. 38(2):119–134.Crossref, Google Scholar
• Peng S (1993) Backward stochastic differential equations and applications to optimal control. Appl. Math. Optim. 27(2):125–144.Crossref, Google Scholar
• Peng S (1997) BSDE and stochastic optimizations. Topics in Stochastic Anal. (Science Press, Beijing).Google Scholar
• Peng S (2010) Nonlinear expectations and stochastic calculus under uncertainty. arXiv preprint arXiv:1002.4546v1.Google Scholar
• Pham T, Zhang J (2014) Two person zero-sum game in weak formulation and path dependent Bellman-Isaacs equation. SIAM J. Control Optim. 52(4):2090–2121.Crossref, Google Scholar
• Sirbu M (2014) Stochastic Perron’s method and elementary strategies for zero-sum differential games. SIAM J. Control Optim. 52(3):1693–1711.Crossref, Google Scholar
• Soner HM, Touzi N (2002) Dynamic programming for stochastic target problems and geometric flows. J. Eur. Math. Soc. (JEMS) 4(3):201–236.Crossref, Google Scholar
• Soner HM, Touzi N (2002) Stochastic target problems, dynamic programming, and viscosity solutions. SIAM J. Control Optim. 41(2):404–424.Crossref, Google Scholar
• Soner HM, Touzi N, Zhang J (2011) Martingale representation theorem for the G-expectation. Stochastic Process. Appl. 121(2):265–287.Crossref, Google Scholar
• Soner HM, Touzi N, Zhang J (2012) Wellposedness of second order backward SDEs. Probab. Theory Related Fields 153(1–2):149–190.Crossref, Google Scholar
• Soner HM, Touzi N, Zhang J (2013) Dual formulation of second order target problems. Ann. Appl. Probab. 23(1):308–347.Crossref, Google Scholar
• Talay D, Zheng Z (2002) Worst case model risk management. Finance Stoch. 6(4):517–537.Crossref, Google Scholar
• Tevzadze R, Toronjadze T, Uzunashvili T (2013) Robust utility maximization for a diffusion market model with misspecified coefficients. Finance Stoch. 17(3):535–563.Crossref, Google Scholar