Probabilistic interpretation for solutions of fully nonlinear stochastic PDEs

Aman, A., Mrhardy, N.: Obstacle problem for SPDE with onlinear Neumann boundary condition via reflected generalized backward doubly SDEs. Stat. Probab. Lett. 83(3), 863–874 (2013) Avellaneda, M., Levy, A., Paras, A.: Pricing and hedging derivative securities in markets with uncertain volatility. Appl. Math. Finance 2(2), 73–88 (1995) Bachouch, A., Gobet, E., Matoussi, A.: Empirical regression method for backward doubly stochastic differential equations. SIAM/ASA J. Uncertain. Quantif. 4(1), 358–379 (2016) Bachouch, A., Lasmar, A.B., Matoussi, A., Mnif, M.: Numerical scheme for semilinear SPDEs via backward doubly SDEs. Stoch. Partial Differ. Equ.: Anal. Comput. 1, 1–43 (2016) Bally, V., Matoussi, A.: Weak solutions for SPDEs and backward doubly stochastic differential equations. J. Theor. Probab. 14(1), 125–164 (2001) Bertsekas, D., Shreve, S.: Stochastic Optimal Control: The Discrete-time Case. Academic Press, New York (1978) Bichteler, K.: Stochastic integration and \(L^{p}-\)theory of semimartingales. Ann. Probab. 9(1), 49–89 (1981) Buckdahn, R., Bulla, I., Ma, J.: Pathwise Taylor expansions for Itō random fields. Math. Control Relat. Fields 1(4), 437–468 (2011) Buckdahn, R., Ma, J.: Stochastic viscosity solutions for nonlinear stochastic partial differential equations. I. Stoch. Process. Appl. 93(2), 181–204 (2001) Buckdahn, R., Ma, J.: Stochastic viscosity solutions for nonlinear stochastic partial differential equations. II. Stoch. Process. Appl. 93(2), 205–228 (2001) Buckdahn, R., Ma, J.: Pathwise stochastic Taylor expansions and stochastic viscosity solutions for fully nonlinear stochastic PDEs. Ann. Probab. 30(3), 1131–1171 (2002) Buckdahn, R., Ma, J.: Pathwise stochastic control problems and stochastic HJB equations. SIAM J. Control Optim. 45(6), 2224–2256 (2007) Buckdahn, R., Ma, J., Zhang, J.: Pathwise Taylor expansions for random fields on multiple dimensional paths. Stoch. Process. Appl. 125(7), 2820–2855 (2015) Buckdahn, R., Ma, J., Zhang, J.: Pathwise viscosity solutions of stochastic PDEs and forward path-dependent PDEs. (2015). arXiv:1501.06978 Caruana, M., Friz, P., Oberhauser, H.: A (rough) pathwise approach to a class of non-linear stochastic partial differential equations. Ann. de l’institut Henri Poincaré Anal. Non Linéaire (C) 28(1), 27–46 (2011) Chen, Z., Peng, S.: A general downcrossing inequality for \(g\)-martingales. Stat. Probab. Lett. 46(2), 169–175 (2000) Cheridito, P., Soner, H., Touzi, N., Victoir, N.: Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs. Commun. Pure Appl. Math. 60(7), 1081–1110 (2007) Crandall, M., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27(1), 1–67 (1992) Dalang, R., Khoshnevisan, D., Nualart, E.: Hitting probabilities for systems of non-linear stochastic heat equations with additive noise. ALEA Latin Am. J. Probab. Math. Stat. 3, 231–271 (2007) Dawson, D.: Stochastic evolution equations. Math. Biosci. 15(3), 287–316 (1972) Dellacherie, C., Meyer, P.: Probabilités et Potentiel, Chapitres XII à XVI, Théorie du Potentiel. Hermann, Paris (1980) Denis, L., Martini, C.: A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab. 16(2), 827–852 (2006) Diehl, J., Friz, P.: Backward stochastic differential equations with rough drivers. Ann. Probab. 40(4), 1715–1758 (2012) Doob, J.L.: Classical potential theory and its probabilistic counterpart. Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1984 edition El Karoui, N., Hamadène, S., Matoussi, A.: Backward stochastic differential equations and applications. Chapter 8 in the book Indifference Pricing: Theory and Applications, pp. 267–320. Springer, New York (2008) El Karoui, N., Tan, X.: Capacities, measurable selection and dynamic programming part I: abstract framework. (2013). arXiv:1310.3363 El Karoui, N., Tan, X.: Capacities, measurable selection and dynamic programming part II: application in stochastic control problems. (2013). arXiv:1310.3364 Fremlin, D.H.: Consequences of Martin’s Axiom, vol. 84 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1984) Friz, P., Gassiat, P., Lions, P.-L., Souganidis, P.: Eikonal equations and pathwise solutions to fully non-linear SPDEs. (2016). arXiv:1602.04746 Friz, P.K., Gassiat, P., Lions, P.-L., Souganidis, P.E.: Eikonal equations and pathwise solutions to fully non-linear spdes. Stoch. Partial Differ. Equ.: Anal. Comput. 5(2), 256–277 (2017) Gerencsér, M., Gyöngy, I., Krylov, N.: On the solvability of degenerate stochastic partial differential equations in Sobolev spaces. Stoch. Partial Differ. Equ. Anal. Comput. 3(1), 52–83 (2015) Gubinelli, M., Tindel, S., Torrecilla, I.: Controlled viscosity solutions of fully nonlinear rough PDEs. (2014). arXiv:1403.2832 Gyöngy, I., Krylov, N.: Accelerated finite difference schemes for linear stochastic partial differential equations in the whole space. SIAM J. Math. Anal. 42(5), 2275–2296 (2010) Gyöngy, I., Krylov, N.: Accelerated numerical schemes for PDEs and SPDEs. In: Stochastic Analysis 2010. Springer, Heidelberg, pp. 131–168 (2011) Hamadène, S., Ouknine, Y.: Reflected backward sdes with general jumps. Theory Probab. Appl. 60(2), 357–376 (2015) Ichikawa, A.: Linear stochastic evolution equations in Hilbert space. J. Differ. Equ. 28(2), 266–277 (1978) Karandikar, R.: On pathwise stochastic integration. Stoch. Process. Appl. 57, 11–18 (1995) Kazi-Tani, N., Possamaï, D., Zhou, C.: Second order BSDEs with jumps: existence and probabilistic representation for fully-nonlinear PIDEs. Electron. J. Probab. 20 (2015) Krylov, N., Rozovskiĭ, B.: On the Cauchy problem for linear stochastic partial differential equations. Izv.: Math. 11(6), 1267–1284 (1977) Krylov, N., Rozovskiĭ, B.: Stochastic evolution equations. J. Soviet Math. 16(4), 1233–1277 (1981) Kunita, H.: Stochastic Flows and Stochastic Differential Equations, vol. 24 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1990) Lin, Y.: A new existence result for second-order BSDEs with quadratic growth and their applications. Stoch.: Int. J. Probab. Stoch. Process. 88(1), 128–146 (2016) Lions, P.-L., Souganidis, P.E.: Fully nonlinear stochastic partial differential equations: non-smooth equations and applications. Comptes Rendus de l’Acad. des Sci.-Ser. I-Math. 327(8), 735–741 (1998) Lions, P.-L., Souganidis, P.E.: Fully nonlinear viscosity stochastic partial differential equations: non-smooth equations and applications. CR Acad. Sci. Paris 327(1), 735–741 (1998) Lions, P.-L., Souganidis, P.E.: Équations aux dérivées partielles stochastiques nonlinéaires et solutions de viscosité. Séminaire équations aux dérivées partielles 1998–1999(1), 1–13 (2000) Lions, P.-L., Souganidis, P.E.: Viscosity solutions of fully nonlinear stochastic partial differential equations. Sūrikaisekikenkyūsho Kōkyūroku, 1287, 58–65. (2002). Viscosity solutions of differential equations and related topics (Japanese) (Kyoto, 2001) Lyons, T.J.: Uncertain volatility and the risk-free synthesis of derivatives. Appl. Math. Finance 2, 117–133 (1995) Ma, J., Wu, Z., Zhang, D., Zhang, J., et al.: On well-posedness of forward-backward sdes: a unified approach. Ann. Appl. Probab. 25(4), 2168–2214 (2015) Matoussi, A., Sheutzow, M.: Semilinear stochastic PDE’s with nonlinear noise and backward doubly SDE’s. J. Theor. Probab. 15, 1–39 (2002) Nutz, M.: Pathwise construction of stochastic integrals. Electron. Commun. Probab. 17(24), 1–7 (2012) Nutz, M.: A quasi-sure approach to the control of non-Markovian stochastic differential equations. Electron. J. Probab. 17(23), 1–23 (2012) Ocone, D., Pardoux, E.: A generalized itô–ventzell formula. application to a class of anticipating stochastic differential equations. Ann. de l’institut Henri Poincaré Probab. et Stat. (B) 25(1), 39–71 (1989) Pardoux, É.: Stochastic partial differential equations and filtering of diffusion processes. Stochastics 3(1–4), 127–167 (1980) Pardoux, É., Peng, S.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14(1), 55–61 (1990) Pardoux, É., Peng, S.: Backward doubly sde’s and systems of quasilinear spdes. Probab. Theory Relat. Field 98, 209–227 (1994) Pardoux, É., Protter, P.: A two-sided stochastic integral and its calculus. Probab. Theory Relat. Field 76(1), 15–49 (1987) Peng, S.: Backward SDE and related \(g-\)expectation. In: El Karoui, N., Mazliak, L. (eds.) Backward Stochastic Differential Equations, vol. 364 of Pitman Research Notes in Mathematics, pp. 141–159. Longman, Harlow (1997) Peng, S.: Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob–Meyer’s type. Probab. Theory Relat. Fields 113(4), 473–499 (1999) Peng, S., Shi, Y.: A type of time-symmetric forward-backward stochastic differential equations. C.R. Math. 336(9), 773–778 (2003) Possamaï, D.: Second order backward stochastic differential equations under a monotonicity condition. Stoch. Process. Appl. 123(5), 1521–1545 (2013) Possamaï, D., Tan, X.: Weak approximation of second-order BSDEs. Ann. Appl. Probab. 25(5), 2535–2562 (2015) Possamaï, D., Tan, X., Zhou, C.: Stochastic control for a class of non–linear stochastic kernels and applications. (2015). arXiv:1510.08439 Possamaï, D., Zhou, C.: Second order backward stochastic differential equations with quadratic growth. Stoch. Process. Appl. 123(10), 3770–3799 (2013) Ren, Z., Tan, X.: On the convergence of monotone schemes for path—dependent PDE. (2015). arXiv:1504.01872 Shi, Y., Gu, Y., Liu, K.: Comparison theorems of backward doubly stochastic differential equations and applications. Stoch. Anal. Appl. 23(1), 97–110 (2005) Soner, H., Touzi, N., Zhang, J.: Martingale representation theorem for the \(G\)-expectation. Stoch. Process. Appl. 121(2), 265–287 (2011) Soner, H., Touzi, N., Zhang, J.: Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16(67), 1844–1879 (2011) Soner, H., Touzi, N., Zhang, J.: Dual formulation of second order target problems. Ann. Appl. Probab. 23(1), 308–347 (2013) Soner, H.M., Touzi, N., Zhang, J.: Wellposedness of second order backward SDEs. Probab. Theory Relat. Fields 153(1–2), 149–190 (2012) Stricker, C., Yor, M.: Calcul stochastique dépendant d’un paramètre. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 45(2), 109–133 (1978) Stroock, D., Varadhan, S.: Multidimensional Diffusion Processes. Springer, Berlin (1979)