UNIQUENESS OF VISCOSITY SOLUTIONS OF STOCHASTIC HAMILTON-JACOBI EQUATIONS
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摘要
This article is devoted to the study of fully nonlinear stochastic Hamilton-Jacobi(HJ) equations for the optimal stochastic control problem of ordinary differential equations with random coefficients. Under the standard Lipschitz continuity assumptions on the coefficients, the value function is proved to be the unique viscosity solution of the associated stochastic HJ equation.
This article is devoted to the study of fully nonlinear stochastic Hamilton-Jacobi(HJ) equations for the optimal stochastic control problem of ordinary differential equations with random coefficients. Under the standard Lipschitz continuity assumptions on the coefficients, the value function is proved to be the unique viscosity solution of the associated stochastic HJ equation.
This article is devoted to the study of fully nonlinear stochastic Hamilton-Jacobi(HJ) equations for the optimal stochastic control problem of ordinary differential equations with random coefficients. Under the standard Lipschitz continuity assumptions on the coefficients, the value function is proved to be the unique viscosity solution of the associated stochastic HJ equation.
引文
[1]Briand P,Delyon B,Hu Y,Pardoux E,Stoica L.Lpsolutions of backward stochastic differential equations.Stoch Process Appl,2003,108(4):604-618
[2]Buckdahn R,Keller C,Ma J,Zhang J.Pathwise viscosity solutions of stochastic PDEs and forward pathdependent PDEs-a rough path view.arXiv:1501.06978,2015
[3]Buckdahn R,Ma J.Pathwise stochastic control problems and stochastic HJB equations.SIAM J Control Optim,2007,45(6):2224-2256
[4]Cont R,Fournié D-A,et al.Functional it? calculus and stochastic integral representation of martingales.The Annals of Probability,2013,41(1):109-133
[5]Crandall M G,Ishii H,Lions P-L.Users guide to viscosity solutions of second order partial differential equations.Bull Amer Math Soc,1992,27(1):1-67
[6]Crandall M G,Kocan M,′Swiech A.Lp-theory for fully nonlinear uniformly parabolic equations.Commun Partial Differ Equ,2000,25(11/12):1997-2053
[7]Da Prato G,Zabczyk J.Stochastic equations in infinite dimensions.Cambridge University Press,2014
[8]Du K,Qiu J,Tang S.Lptheory for super-parabolic backward stochastic partial differential equations in the whole space.Appl Math Optim,2011,65(2):175-219
[9]Ekren I,Keller C,Touzi N,Zhang J.On viscosity solutions of path dependent PDEs.Ann Probab,2014,42(1):204-236
[10]Ekren I,Touzi N,Zhang J.Viscosity solutions of fully nonlinear parabolic path dependent PDEs:Part I.Ann Probab,2016,44(2):1212-1253
[11]Gawarecki L,Mandrekar V.Stochastic differential equations in infinite dimensions:with applications to stochastic partial differential equations.Springer Science&Business Media,2010
[12]Horst U,Qiu J,Zhang Q.A constrained control problem with degenerate coefficients and degenerate backward SPDEs with singular terminal condition.SIAM J Control Optim,2016,54(2):946-963
[13]Hu Y,Ma J,Yong J.On semi-linear degenerate backward stochastic partial differential equations.Probab Theory Relat Fields,2002,123:381-411
[14]Juutinen P.On the definition of viscosity solutions for parabolic equations.Proceedings of the American Mathematical Society,2001,129(10):2907-2911
[15]Krylov N V.Boundedly nonhomogeneous elliptic and parabolic equations.Izvestiya Rossiiskoi Akademii Nauk.Seriya Matematicheskaya,1982,46(3):487-523
[16]Krylov N V.Nonlinear Elliptic and Parabolic Equations of the Second Order.Dordrecht:D Reidel,1987
[17]Le?ao D,Ohashi A,Simas A.A weak version of path-dependent functional It? calculus.Ann Probab,2018,46(6):3399-3441
[18]Lions P,Souganidis P.Fully nonlinear stochastic partial differential equations:Non-smooth equations and applications.C R Acad Sci paris,1998,327(1):735-741
[19]Lions P L.Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations.Part II.Commun Partial Differ Equ,1983,8:1229-1276
[20]Lukoyanov N Y.On viscosity solution of functional hamilton-jacobi type equations for hereditary systems.Proceedings of the Steklov Institute of Mathematics,2007,259(2):S190-S200
[21]Ma J,Yin H,Zhang J.On non-Markovian forward-backward SDEs and backward stochastic PDEs.Stoch Process Appl,2012,122(12):3980-4004
[22]?ksendal B.Stochastic differential equations.Springer,2003
[23]Pardoux E.Stochastic partial differential equations and filtering of diffusion processes.Stoch,1979:127-167
[24]Peng S.Stochastic Hamilton-Jacobi-Bellman equations.SIAM J Control Optim,1992,30:284-304
[25]Peng S.Backward stochastic differential equation,nonlinear expectation and their applications.Proceedings of the International Congress of Mathematicians,2010:393-432
[26]Peng S.Note on viscosity solution of path-dependent PDE and G-martingales.arXiv:1106.1144,2011
[27]Qiu J.Weak solution for a class of fully nonlinear stochastic hamilton-jacobi-bellman equations.Stoch Process Appl,2017,127(6):1926-1959
[28]Qiu J.Viscosity solutions of stochastic Hamilton-Jacobi-Bellman equations.SIAM J Control Optim,2018,56(5):3708-3730
[29]Qiu J,Tang S.Maximum principles for backward stochastic partial differential equations.J Funct Anal,2012,262:2436-2480
[30]Tang S.Semi-linear systems of backward stochastic partial differential equations in Rn.Chin Ann Math,2005,26B(3):437-456
[31]Tang S,Wei W.On the cauchy problem for backward stochastic partial differential equations in H?lder spaces.Ann Probab,2016,44(1):360-398
[32]Wang L.On the regularity theory of fully nonlinear parabolic equations:I.Commun Pure Appl Math,1992,45(1):27-76
1Recall thatτT denotes the set of stopping times ζ satifying τ≤ζ≤T as defined in Section 2.2.
2As U?Rn is a nonempty compact set, it has a denumerable subset K?U that is dense in U, and by the continuity of the coefficients, the essential infimum may be taken overK. This together with some basic properties of viscosity solutions(see[32, Proposition 3.7] for instance)allows[15, Theorem 1.1] to be applied straightforwardly.
[2]Buckdahn R,Keller C,Ma J,Zhang J.Pathwise viscosity solutions of stochastic PDEs and forward pathdependent PDEs-a rough path view.arXiv:1501.06978,2015
[3]Buckdahn R,Ma J.Pathwise stochastic control problems and stochastic HJB equations.SIAM J Control Optim,2007,45(6):2224-2256
[4]Cont R,Fournié D-A,et al.Functional it? calculus and stochastic integral representation of martingales.The Annals of Probability,2013,41(1):109-133
[5]Crandall M G,Ishii H,Lions P-L.Users guide to viscosity solutions of second order partial differential equations.Bull Amer Math Soc,1992,27(1):1-67
[6]Crandall M G,Kocan M,′Swiech A.Lp-theory for fully nonlinear uniformly parabolic equations.Commun Partial Differ Equ,2000,25(11/12):1997-2053
[7]Da Prato G,Zabczyk J.Stochastic equations in infinite dimensions.Cambridge University Press,2014
[8]Du K,Qiu J,Tang S.Lptheory for super-parabolic backward stochastic partial differential equations in the whole space.Appl Math Optim,2011,65(2):175-219
[9]Ekren I,Keller C,Touzi N,Zhang J.On viscosity solutions of path dependent PDEs.Ann Probab,2014,42(1):204-236
[10]Ekren I,Touzi N,Zhang J.Viscosity solutions of fully nonlinear parabolic path dependent PDEs:Part I.Ann Probab,2016,44(2):1212-1253
[11]Gawarecki L,Mandrekar V.Stochastic differential equations in infinite dimensions:with applications to stochastic partial differential equations.Springer Science&Business Media,2010
[12]Horst U,Qiu J,Zhang Q.A constrained control problem with degenerate coefficients and degenerate backward SPDEs with singular terminal condition.SIAM J Control Optim,2016,54(2):946-963
[13]Hu Y,Ma J,Yong J.On semi-linear degenerate backward stochastic partial differential equations.Probab Theory Relat Fields,2002,123:381-411
[14]Juutinen P.On the definition of viscosity solutions for parabolic equations.Proceedings of the American Mathematical Society,2001,129(10):2907-2911
[15]Krylov N V.Boundedly nonhomogeneous elliptic and parabolic equations.Izvestiya Rossiiskoi Akademii Nauk.Seriya Matematicheskaya,1982,46(3):487-523
[16]Krylov N V.Nonlinear Elliptic and Parabolic Equations of the Second Order.Dordrecht:D Reidel,1987
[17]Le?ao D,Ohashi A,Simas A.A weak version of path-dependent functional It? calculus.Ann Probab,2018,46(6):3399-3441
[18]Lions P,Souganidis P.Fully nonlinear stochastic partial differential equations:Non-smooth equations and applications.C R Acad Sci paris,1998,327(1):735-741
[19]Lions P L.Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations.Part II.Commun Partial Differ Equ,1983,8:1229-1276
[20]Lukoyanov N Y.On viscosity solution of functional hamilton-jacobi type equations for hereditary systems.Proceedings of the Steklov Institute of Mathematics,2007,259(2):S190-S200
[21]Ma J,Yin H,Zhang J.On non-Markovian forward-backward SDEs and backward stochastic PDEs.Stoch Process Appl,2012,122(12):3980-4004
[22]?ksendal B.Stochastic differential equations.Springer,2003
[23]Pardoux E.Stochastic partial differential equations and filtering of diffusion processes.Stoch,1979:127-167
[24]Peng S.Stochastic Hamilton-Jacobi-Bellman equations.SIAM J Control Optim,1992,30:284-304
[25]Peng S.Backward stochastic differential equation,nonlinear expectation and their applications.Proceedings of the International Congress of Mathematicians,2010:393-432
[26]Peng S.Note on viscosity solution of path-dependent PDE and G-martingales.arXiv:1106.1144,2011
[27]Qiu J.Weak solution for a class of fully nonlinear stochastic hamilton-jacobi-bellman equations.Stoch Process Appl,2017,127(6):1926-1959
[28]Qiu J.Viscosity solutions of stochastic Hamilton-Jacobi-Bellman equations.SIAM J Control Optim,2018,56(5):3708-3730
[29]Qiu J,Tang S.Maximum principles for backward stochastic partial differential equations.J Funct Anal,2012,262:2436-2480
[30]Tang S.Semi-linear systems of backward stochastic partial differential equations in Rn.Chin Ann Math,2005,26B(3):437-456
[31]Tang S,Wei W.On the cauchy problem for backward stochastic partial differential equations in H?lder spaces.Ann Probab,2016,44(1):360-398
[32]Wang L.On the regularity theory of fully nonlinear parabolic equations:I.Commun Pure Appl Math,1992,45(1):27-76
1Recall thatτT denotes the set of stopping times ζ satifying τ≤ζ≤T as defined in Section 2.2.
2As U?Rn is a nonempty compact set, it has a denumerable subset K?U that is dense in U, and by the continuity of the coefficients, the essential infimum may be taken overK. This together with some basic properties of viscosity solutions(see[32, Proposition 3.7] for instance)allows[15, Theorem 1.1] to be applied straightforwardly.