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Fully nonlinear stochastic and rough PDEs: Classical and viscosity solutions
1. Univ Brest, UMR CNRS 6205, Laboratoire de Mathématiques de Bretagne Atlantique, Brest, France, and Shandong University, Jinan, China |
2. Department of Mathematics, University of Central Florida, Orlando, Florida, United States |
3. Department of Mathematics, University of Southern California, Los Angeles, California, United States |
References:
[1] |
Buckdahn, R., I. Bulla, and J. Ma. (2011). On Pathwise Stochastic Taylor Expansions, Math. Control Relat. Fields 1, no. 4, 437-468. |
[2] |
Buckdahn, R. and J. Li. (2008). Stochastic differential games and viscosity solutions of Hamilton-JacobiBellman-Isaacs equations, SIAM J. Control Optim. 47, no. 1, 444-475. |
[3] |
Buckdahn, R. and J. Ma. (2001). Stochastic viscosity solutions for nonlinear stochastic partial differential equations. I, Stoch. Process. Appl. 93, no. 2, 181-204. |
[4] |
Buckdahn, R. and J. Ma. (2001). Stochastic viscosity solutions for nonlinear stochastic partial differential equations. II, Stoch. Process. Appl. 93, no. 2, 205-228. |
[5] |
Buckdahn, R. and J. Ma. (2002). Pathwise stochastic Taylor expansions and stochastic viscosity solutions for fully nonlinear stochastic PDEs, Ann. Probab. 30, no. 3, 1131-1171. |
[6] |
Buckdahn, R. and J. Ma. (2007). Pathwise stochastic control problems and stochastic HJB equations, SIAM J. Control Optim. 45, no. 6, 2224-2256. |
[7] |
Buckdahn, R., J. Ma, and J. Zhang. (2015). Pathwise Taylor expansions for random fields on multiple dimensional paths, Stoch. Process. Appl. 125, 2820-2855. |
[8] |
Caruana, M., P. Friz, and H. Oberhauser. (2011). A (rough) pathwise approach to a class of non-linear stochastic partial differential equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 28, 27-46. |
[9] |
Crandall, M.G., H. Ishii, and P.-L. Lions. (1992). User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27, no. 1, 1-67. |
[10] |
Da Prato, G. and L. Tubaro. (1996). Fully nonlinear stochastic partial differential equations, SIAM J. Math. Anal. 27, no. 1, 40-55. |
[11] |
Davis, M. and G. Burstein. (1992). A Deterministic Approach To Stochastic Optimal Control With Application To Anticipative Control, Stochast. Stoch. Rep. 40, no. 3-4, 203-256. |
[12] |
Diehl, J. and P. Friz. (2012). Backward stochastic differential equations with rough drivers, Ann. Prob. 40, 1715-1758. |
[13] |
Diehl, J., P. Friz, and P. Gassiat. (2017). Stochastic control with rough paths, Appl. Math. Optim. 75, no. 2, 285-315. |
[14] |
Diehl, J., P. Friz, and H. Oberhauser. (2014). Regularity theory for rough partial differential equations and parabolic comparison revisited, Springer, Cham. |
[15] |
Diehl, J., H. Oberhauser, and S. Riedel. (2015). A Lévy area between Brownian motion and rough paths with applications to robust nonlinear filtering and rough partial differential equations, Stoch. Process. Appl. 125, no. 1, 161-181. |
[16] |
Dupire, B. (2019). Functional Itô calculus, Quant. Finan. 19, no. 5, 721-729. |
[17] |
Ekren, I., C. Keller, N. Touzi, and J. Zhang. (2014). On viscosity solutions of path dependent PDEs, Ann. Probab. 42, 204-236. |
[18] |
Ekren, I., N. Touzi, and J. Zhang. (2016). Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs:Part I, Ann. Probab. 44, 1212-1253. |
[19] |
Ekren, I., N. Touzi, and J. Zhang. (2016). Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs:Part II, Ann. Probab. 44, 2507-2553. |
[20] |
Friz, P., P. Gassiat, P.L. Lions, and P.E. Souganidis. (2017). Eikonal equations and pathwise solutions to fully non-linear SPDEs, Stochast. Partial Differ. Equ. Anal. Comput. 5, 256-277. |
[21] |
Friz, P. and M. Hairer. (2014). A course on rough paths:With an introduction to regularity structures, Universitext, Springer, Cham. |
[22] |
Friz, P. and H. Oberhauser. (2011). On the splitting-up method for rough (partial) differential equations, J. Differ. Equ. 251, no. 2, 316-338. |
[23] |
Friz, P. and H. Oberhauser. (2014). Rough path stability of (semi-)linear SPDEs, Probab. Theory Relat. Fields 158, 401-434. |
[24] |
Gilbarg, D. and N. Trudinger. (1983). Elliptic Partial Differential Equations of second order, second edition, Springer-Verlag, Germany. |
[25] |
Gubinelli, M. (2004). Controlling rough paths, J. Funct. Anal. 216, no. 1, 86-140. |
[26] |
Gubinelli, M., S. Tindel, and I. Torrecilla. (2014). Controlled viscosity solutions of fully nonlinear rough PDEs. arXiv preprint, arXiv:1403.2832. |
[27] |
Keller, C. and J. Zhang. (2016). Pathwise Itô calculus for rough paths and rough PDEs with path dependent coefficients, Stoch. Process. Appl. 126, 735-766. |
[28] |
Krylov, N.V. (1999). An analytic approach to SPDEs, Stoch. Partial Differ. Equ. Six Perspect. Math. Surv. Monogr. Amer. Math. Soc. Providence RI 64, 185-242. |
[29] |
Kunita, H. (1997). Stochastic flows and stochastic differential equations, Cambridge University Press, Cambridge. |
[30] |
Lieberman, G. (1996). Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge. |
[31] |
Lions, P.-L. and P. E. Souganidis. (1998). Fully nonlinear stochastic partial differential equations, C. R. Acad. Sci. Paris Sér. I Math. 326, no. 9, 1085-1092. |
[32] |
Lions, P.-L. and P. E. Souganidis. (1998). Fully nonlinear stochastic partial differential equations:nonsmooth equations and applications, C. R. Acad. Sci. Paris Sér. I Math. 327, no. 8, 735-741. |
[33] |
Lions, P.-L. and P. E. Souganidis. (2000). Fully nonlinear stochastic PDE with semilinear stochastic dependence, C. R. Acad. Sci. Paris Sér. I Math. 331, no. 8, 617-624. |
[34] |
Lions, P.-L. and P.E. Souganidis. (2000). Uniqueness of weak solutions of fully nonlinear stochastic partial differential equations, C. R. l'Acad. Sci.-Ser. I-Math. 331, no. 10, 783-790. |
[35] |
Lunardi, A. (1995). Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and their Applications 16, Birkhäuser Verlag, Basel. |
[36] |
Lyons, T. (1998). Differential equations driven by rough signals, Rev. Mat. Iberoam. 14, no. 2, 215-310. |
[37] |
Matoussi, A., D. Possamai, and W. Sabbagh. (2018). Probabilistic interpretation for solutions of Fully Nonlinear Stochastic PDEs, Probab. Theory Relat. Fields. https://doi.org/10.1007/s00440-018-0859-4. |
[38] |
Mikulevicius, R. and G. Pragarauskas. (1994). Classical solutions of boundary value problems for some nonlinear integro-differential equations, Lithuanian Math. J. 34, no. 3, 275-287. |
[39] |
Musiela, M. and T. Zariphopoulou. (2010). Stochastic partial differential equations and portfolio choice, Contemporary Quantitative Finance, Springer, Berlin. |
[40] |
Nadirashvili, N. and S. Vladut. (2007). Nonclassical solutions of fully nonlinear elliptic equations, Geom. Funct. Anal. 17, no. 4, 1283-1296. |
[41] |
Pardoux, E. and S. Peng. (1994). Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probab. Theory Relat. Fields 98, 209-227. |
[42] |
Peng, S. (1992). Stochastic Hamilton-Jacobi-Bellman equations, SIAM J. Control Optim 30, no. 2, 284- 304. |
[43] |
Pham, T. and J. Zhang. (2014). Two Person Zero-sum Game in Weak Formulation and Path Dependent Bellman-Isaacs Equation, SIAM J. Control. Optim. 52, 2090-2121. |
[44] |
Rozovskii, B.L. (1990). Stochastic Evolution Systems:Linear Theory and Applications to Non-linear Filtering, Kluwer Academic Publishers, Boston. |
[45] |
Safonov, M.V. (1988). Boundary value problems for second-order nonlinear parabolic equations, (Russian), Funct. Numer. Methods Math. Phys. "Naukova Dumka" Kiev. 274, 99-203. |
[46] |
Safonov, M.V. (1989). Classical solution of second-order nonlinear elliptic equations, Math. USSR-Izv 33, no. 3, 597-612. |
[47] |
Seeger, B. (2018). Perron's method for pathwise viscosity solutions, Commun. Partial Differ. Equ. 43, no. 6, 998-1018. |
[48] |
Seeger, B. (2018). Homogenization of pathwise Hamilton-Jacobi equations, J. Math. Pures Appl. 110, 1-31. |
[49] |
Seeger, B. (2020). Approximation schemes for viscosity solutions of fully nonlinear stochastic partial differential equations, Ann. Appl. Probab. 30, no. 4, 1784-1823. |
[50] |
Souganidis, P.E. (2019). Pathwise solutions for fully nonlinear first- and second-order partial differential equations with multiplicative rough time dependence, Singular random dynamics, Lecture Notes in Math. vol. 2253, Springer, Cham. |
[51] |
Zhang, J. (2017). Backward Stochastic Differential Equations-from linear to fully nonlinear theory, Springer, New York. |
show all references
References:
[1] |
Buckdahn, R., I. Bulla, and J. Ma. (2011). On Pathwise Stochastic Taylor Expansions, Math. Control Relat. Fields 1, no. 4, 437-468. |
[2] |
Buckdahn, R. and J. Li. (2008). Stochastic differential games and viscosity solutions of Hamilton-JacobiBellman-Isaacs equations, SIAM J. Control Optim. 47, no. 1, 444-475. |
[3] |
Buckdahn, R. and J. Ma. (2001). Stochastic viscosity solutions for nonlinear stochastic partial differential equations. I, Stoch. Process. Appl. 93, no. 2, 181-204. |
[4] |
Buckdahn, R. and J. Ma. (2001). Stochastic viscosity solutions for nonlinear stochastic partial differential equations. II, Stoch. Process. Appl. 93, no. 2, 205-228. |
[5] |
Buckdahn, R. and J. Ma. (2002). Pathwise stochastic Taylor expansions and stochastic viscosity solutions for fully nonlinear stochastic PDEs, Ann. Probab. 30, no. 3, 1131-1171. |
[6] |
Buckdahn, R. and J. Ma. (2007). Pathwise stochastic control problems and stochastic HJB equations, SIAM J. Control Optim. 45, no. 6, 2224-2256. |
[7] |
Buckdahn, R., J. Ma, and J. Zhang. (2015). Pathwise Taylor expansions for random fields on multiple dimensional paths, Stoch. Process. Appl. 125, 2820-2855. |
[8] |
Caruana, M., P. Friz, and H. Oberhauser. (2011). A (rough) pathwise approach to a class of non-linear stochastic partial differential equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 28, 27-46. |
[9] |
Crandall, M.G., H. Ishii, and P.-L. Lions. (1992). User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27, no. 1, 1-67. |
[10] |
Da Prato, G. and L. Tubaro. (1996). Fully nonlinear stochastic partial differential equations, SIAM J. Math. Anal. 27, no. 1, 40-55. |
[11] |
Davis, M. and G. Burstein. (1992). A Deterministic Approach To Stochastic Optimal Control With Application To Anticipative Control, Stochast. Stoch. Rep. 40, no. 3-4, 203-256. |
[12] |
Diehl, J. and P. Friz. (2012). Backward stochastic differential equations with rough drivers, Ann. Prob. 40, 1715-1758. |
[13] |
Diehl, J., P. Friz, and P. Gassiat. (2017). Stochastic control with rough paths, Appl. Math. Optim. 75, no. 2, 285-315. |
[14] |
Diehl, J., P. Friz, and H. Oberhauser. (2014). Regularity theory for rough partial differential equations and parabolic comparison revisited, Springer, Cham. |
[15] |
Diehl, J., H. Oberhauser, and S. Riedel. (2015). A Lévy area between Brownian motion and rough paths with applications to robust nonlinear filtering and rough partial differential equations, Stoch. Process. Appl. 125, no. 1, 161-181. |
[16] |
Dupire, B. (2019). Functional Itô calculus, Quant. Finan. 19, no. 5, 721-729. |
[17] |
Ekren, I., C. Keller, N. Touzi, and J. Zhang. (2014). On viscosity solutions of path dependent PDEs, Ann. Probab. 42, 204-236. |
[18] |
Ekren, I., N. Touzi, and J. Zhang. (2016). Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs:Part I, Ann. Probab. 44, 1212-1253. |
[19] |
Ekren, I., N. Touzi, and J. Zhang. (2016). Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs:Part II, Ann. Probab. 44, 2507-2553. |
[20] |
Friz, P., P. Gassiat, P.L. Lions, and P.E. Souganidis. (2017). Eikonal equations and pathwise solutions to fully non-linear SPDEs, Stochast. Partial Differ. Equ. Anal. Comput. 5, 256-277. |
[21] |
Friz, P. and M. Hairer. (2014). A course on rough paths:With an introduction to regularity structures, Universitext, Springer, Cham. |
[22] |
Friz, P. and H. Oberhauser. (2011). On the splitting-up method for rough (partial) differential equations, J. Differ. Equ. 251, no. 2, 316-338. |
[23] |
Friz, P. and H. Oberhauser. (2014). Rough path stability of (semi-)linear SPDEs, Probab. Theory Relat. Fields 158, 401-434. |
[24] |
Gilbarg, D. and N. Trudinger. (1983). Elliptic Partial Differential Equations of second order, second edition, Springer-Verlag, Germany. |
[25] |
Gubinelli, M. (2004). Controlling rough paths, J. Funct. Anal. 216, no. 1, 86-140. |
[26] |
Gubinelli, M., S. Tindel, and I. Torrecilla. (2014). Controlled viscosity solutions of fully nonlinear rough PDEs. arXiv preprint, arXiv:1403.2832. |
[27] |
Keller, C. and J. Zhang. (2016). Pathwise Itô calculus for rough paths and rough PDEs with path dependent coefficients, Stoch. Process. Appl. 126, 735-766. |
[28] |
Krylov, N.V. (1999). An analytic approach to SPDEs, Stoch. Partial Differ. Equ. Six Perspect. Math. Surv. Monogr. Amer. Math. Soc. Providence RI 64, 185-242. |
[29] |
Kunita, H. (1997). Stochastic flows and stochastic differential equations, Cambridge University Press, Cambridge. |
[30] |
Lieberman, G. (1996). Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge. |
[31] |
Lions, P.-L. and P. E. Souganidis. (1998). Fully nonlinear stochastic partial differential equations, C. R. Acad. Sci. Paris Sér. I Math. 326, no. 9, 1085-1092. |
[32] |
Lions, P.-L. and P. E. Souganidis. (1998). Fully nonlinear stochastic partial differential equations:nonsmooth equations and applications, C. R. Acad. Sci. Paris Sér. I Math. 327, no. 8, 735-741. |
[33] |
Lions, P.-L. and P. E. Souganidis. (2000). Fully nonlinear stochastic PDE with semilinear stochastic dependence, C. R. Acad. Sci. Paris Sér. I Math. 331, no. 8, 617-624. |
[34] |
Lions, P.-L. and P.E. Souganidis. (2000). Uniqueness of weak solutions of fully nonlinear stochastic partial differential equations, C. R. l'Acad. Sci.-Ser. I-Math. 331, no. 10, 783-790. |
[35] |
Lunardi, A. (1995). Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and their Applications 16, Birkhäuser Verlag, Basel. |
[36] |
Lyons, T. (1998). Differential equations driven by rough signals, Rev. Mat. Iberoam. 14, no. 2, 215-310. |
[37] |
Matoussi, A., D. Possamai, and W. Sabbagh. (2018). Probabilistic interpretation for solutions of Fully Nonlinear Stochastic PDEs, Probab. Theory Relat. Fields. https://doi.org/10.1007/s00440-018-0859-4. |
[38] |
Mikulevicius, R. and G. Pragarauskas. (1994). Classical solutions of boundary value problems for some nonlinear integro-differential equations, Lithuanian Math. J. 34, no. 3, 275-287. |
[39] |
Musiela, M. and T. Zariphopoulou. (2010). Stochastic partial differential equations and portfolio choice, Contemporary Quantitative Finance, Springer, Berlin. |
[40] |
Nadirashvili, N. and S. Vladut. (2007). Nonclassical solutions of fully nonlinear elliptic equations, Geom. Funct. Anal. 17, no. 4, 1283-1296. |
[41] |
Pardoux, E. and S. Peng. (1994). Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probab. Theory Relat. Fields 98, 209-227. |
[42] |
Peng, S. (1992). Stochastic Hamilton-Jacobi-Bellman equations, SIAM J. Control Optim 30, no. 2, 284- 304. |
[43] |
Pham, T. and J. Zhang. (2014). Two Person Zero-sum Game in Weak Formulation and Path Dependent Bellman-Isaacs Equation, SIAM J. Control. Optim. 52, 2090-2121. |
[44] |
Rozovskii, B.L. (1990). Stochastic Evolution Systems:Linear Theory and Applications to Non-linear Filtering, Kluwer Academic Publishers, Boston. |
[45] |
Safonov, M.V. (1988). Boundary value problems for second-order nonlinear parabolic equations, (Russian), Funct. Numer. Methods Math. Phys. "Naukova Dumka" Kiev. 274, 99-203. |
[46] |
Safonov, M.V. (1989). Classical solution of second-order nonlinear elliptic equations, Math. USSR-Izv 33, no. 3, 597-612. |
[47] |
Seeger, B. (2018). Perron's method for pathwise viscosity solutions, Commun. Partial Differ. Equ. 43, no. 6, 998-1018. |
[48] |
Seeger, B. (2018). Homogenization of pathwise Hamilton-Jacobi equations, J. Math. Pures Appl. 110, 1-31. |
[49] |
Seeger, B. (2020). Approximation schemes for viscosity solutions of fully nonlinear stochastic partial differential equations, Ann. Appl. Probab. 30, no. 4, 1784-1823. |
[50] |
Souganidis, P.E. (2019). Pathwise solutions for fully nonlinear first- and second-order partial differential equations with multiplicative rough time dependence, Singular random dynamics, Lecture Notes in Math. vol. 2253, Springer, Cham. |
[51] |
Zhang, J. (2017). Backward Stochastic Differential Equations-from linear to fully nonlinear theory, Springer, New York. |
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