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Error estimates for second order Hamilton-Jacobi-Bellman equations. Approximation of probabilistic reachable sets
1. | Unité des mathématiques appliquées (UMA), ENSTA ParisTech, 828 Bd Maréchaux, 91120 Palaiseau, France, France |
2. | Laboratoire Jacques-Louis Lions, UMR 7598, Université Paris-Diderot (Paris 7), UFR de Mathématiques - 5 rue Thomas Mann, 75205 Paris CEDEX 13, France |
References:
[1] |
A. Abate, S. Amin, M. Prandini, J. Lygeros and S. Sastry, Computational approaches to reachability analysis of stochastic hybrid systems, Hybrid Systems, 4416 (2007), 4-17.
doi: 10.1007/978-3-540-71493-4_4. |
[2] |
A. Abate, M. Prandini, J. Lygeros and S. Sastry, Probabilistic reachability and safety for controlled discrete time stochastic hybrid systems, Automatica, 44 (2008), 2724-2734.
doi: 10.1016/j.automatica.2008.03.027. |
[3] |
M. Althoff, O. Stursberg and M. Buss, Safety assessement of autonomous cars using verification techniques, American Control Conference, (2007), 4154-4159.
doi: 10.1109/ACC.2007.4282809. |
[4] |
M. Althoff, O. Stursberg and M. Buss, Safety assessement for stochastic linear systems using enclosing hulls of probability density functions, European Control Conference, 625-630. |
[5] |
S. Amin, A. Abate, M. Prandini, S. Sastry and J. Lygeros, Reachability analysis for controlled discrete time stochastic hybrid systems, in Lecture Notes in Computer Science LNCS, 3927 (2006), 49-63.
doi: 10.1007/11730637_7. |
[6] |
G. Barles and E. R. Jakobsen, On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations, ESAIM:M2AN, 36 (2002), 33-54.
doi: 10.1051/m2an:2002002. |
[7] |
G. Barles and E. R. Jakobsen, Error bounds for monotone approximation schemes for Hamilton-Jacobi-Bellman equations, SIAM J. Numer. Anal., 43 (2005), 540-558.
doi: 10.1137/S003614290343815X. |
[8] |
G. Barles and E. R. Jakobsen, Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations, Mathematics of Computations, 76 (2007), 1861-1893.
doi: 10.1090/S0025-5718-07-02000-5. |
[9] |
I. H. Biswas, E. R. Jakobsen and K. H. Karlsen, Difference quadrature schemes for nonlinear degenerate parabolic integro-pde, SIAM J. Numer. Anal., 48 (2010), 1110-1135.
doi: 10.1137/090761501. |
[10] |
I. H. Biswas, E. R. Jakobsen and K. H. Karlsen, Viscosity solutions for a system of integro-pdes and connections to optimal switching and control of jump-diffusion processes, Applied mathematics and optimization, 62 (2010), 47-80.
doi: 10.1007/s00245-009-9095-8. |
[11] |
O. Bokanowski, N. Forcadel and H. Zidani, Reachability and minimal times for state constrained nonlinear problems without any controllability assumption, SIAM J. Control and Optimization. Doi: 10.1137/090762075, 48 (2010), 4292-4316.
doi: 10.1137/090762075. |
[12] |
O. Bokanowski, A. Picarelli and H. Zidani, Dynamic programming and error estimates for stochastic control problems with maximum cost, Applied Math. and Optimization, 71 (2015), 125-163.
doi: 10.1007/s00245-014-9255-3. |
[13] |
J. Bonnans, S. Maroso and H. Zidani, Error bounds for stochastic differential games: The adverse stopping case, IMA, J. Numer. Anal., 26 (2006), 188-212.
doi: 10.1093/imanum/dri034. |
[14] |
J. Bonnans, S. Maroso and H. Zidani, Error estimates for a stochastic impulse control problem, Applied. Math. and Optimisation, 55 (2007), 327-357.
doi: 10.1007/s00245-006-0865-2. |
[15] |
B. Bouchard, R. Elie and N. Touzi, Stochastic target problems with controlled loss, SIAM, 48 (2008), 3123-3150.
doi: 10.1137/08073593X. |
[16] |
A. Briani, F. Camilli and H. Zidani, Approximation schemes for monotone systems of nonlinear second order partial differential equations: convergence result and error estimate, Differential Equations and Applications, 4 (2012), 297-317.
doi: 10.7153/dea-04-18. |
[17] |
L. Caffarelli and P. E. Souganidis, A rate of convergence for monotone finite difference approximations to fully nonlinear uniformly elliptic pde, Comm. Pure Appl. Math., 61 (2008), 1-17.
doi: 10.1002/cpa.20208. |
[18] |
F. Camilli and E. R. Jakobsen, A finite element like scheme for integro-partial differential Hamilton-Jacobi-Bellman equations, SIAM J. Numer. Anal., 47 (2009), 2407-2431.
doi: 10.1137/080723144. |
[19] |
F. Camilli and M. Falcone, An approximation scheme for the optimal control of diffusion processes, RAIRO Modél. Math. Anal. Numér., 29 (1995), 97-122. |
[20] |
F. Da Lio and O. Ley, Uniqueness results for convex Hamilton-Jacobi equations under $p>1$ growth conditions on data, Applied Math. and Optimization, 63 (2011), 309-339. |
[21] |
K. Debrabant and E. R. Jakobsen, Semi-Lagrangian schemes for linear and fully non-linear diffusion equations, Mathematics of Computations, 82 (2013), 1433-1462.
doi: 10.1090/S0025-5718-2012-02632-9. |
[22] |
F. Delarue and S. Menozzi, Density estimates for a random noise propagating through a chain of differential equations, J. Funct. Anal., 259 (2010), 1577-1630.
doi: 10.1016/j.jfa.2010.05.002. |
[23] |
W. H. Fleming and M. H. Soner, Controlled Markov Processes and Viscosity Solutions, vol. 25 of Stochastic Modelling and Applied Probability, 2nd edition, Springer, New York, 2006. |
[24] |
H. Föllmer and P. Leukert, Quantile hedging, SIAM, 3 (1999), 251-273.
doi: 10.1007/s007800050062. |
[25] |
D. Goreac and O.-S. Serea, Mayer and optimal stopping stochastic control problems with discontinuous cost, J. Math. Anal. Appl., 380 (2011), 327-342.
doi: 10.1016/j.jmaa.2011.02.039. |
[26] |
N. V. Krylov, Mean value theorems for stochastic integrals, Ann. Probab., 29 (2001), 385-410.
doi: 10.1214/aop/1008956335. |
[27] |
N. Krylov, On the rate of convergence of finite difference approximation for Bellman's equation, St. Petersburg Math. J., 9 (1998), 639-650. |
[28] |
N. Krylov, On the rate of convergence of finite-difference approximations for Bellman's equations with variable coefficients, Probability Theory and Related Fields, 117 (2000), 1-16.
doi: 10.1007/s004400050264. |
[29] |
N. Krylov, On the rate of convergence for finite-difference approximations for bellman equations with lipschitz coefficients, Applied Mathematics and Optimization, 52 (2005), 365-399.
doi: 10.1007/s00245-005-0832-3. |
[30] |
H. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, vol. 24 of Applications of mathematics, Springer, New York, 2001, Second edition.
doi: 10.1007/978-1-4613-0007-6. |
[31] |
I. Mitchell, A. Bayen and C. Tomlin, A time-dependent Hamiliton-Jacobi formulation of reachable sets for continuous dynamic games, IEEE Transactions on automatic control, 50 (2005), 947-957.
doi: 10.1109/TAC.2005.851439. |
[32] |
R. Munos and H. Zidani, Consistency of a simple multidimensional scheme for hjb equations, C. R. Acad. Sci. Paris, Ser. I, 340 (2005), 499-502.
doi: 10.1016/j.crma.2005.02.001. |
[33] |
S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49.
doi: 10.1016/0021-9991(88)90002-2. |
[34] |
R. Rubinstein and D. Kroese, Simulation and the Monte Carlo Method, Wiley, 2008, Second edition. |
[35] |
J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Stochastic Modelling and Applied Probability, Springer, Paris, 1999.
doi: 10.1007/978-1-4612-1466-3. |
show all references
References:
[1] |
A. Abate, S. Amin, M. Prandini, J. Lygeros and S. Sastry, Computational approaches to reachability analysis of stochastic hybrid systems, Hybrid Systems, 4416 (2007), 4-17.
doi: 10.1007/978-3-540-71493-4_4. |
[2] |
A. Abate, M. Prandini, J. Lygeros and S. Sastry, Probabilistic reachability and safety for controlled discrete time stochastic hybrid systems, Automatica, 44 (2008), 2724-2734.
doi: 10.1016/j.automatica.2008.03.027. |
[3] |
M. Althoff, O. Stursberg and M. Buss, Safety assessement of autonomous cars using verification techniques, American Control Conference, (2007), 4154-4159.
doi: 10.1109/ACC.2007.4282809. |
[4] |
M. Althoff, O. Stursberg and M. Buss, Safety assessement for stochastic linear systems using enclosing hulls of probability density functions, European Control Conference, 625-630. |
[5] |
S. Amin, A. Abate, M. Prandini, S. Sastry and J. Lygeros, Reachability analysis for controlled discrete time stochastic hybrid systems, in Lecture Notes in Computer Science LNCS, 3927 (2006), 49-63.
doi: 10.1007/11730637_7. |
[6] |
G. Barles and E. R. Jakobsen, On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations, ESAIM:M2AN, 36 (2002), 33-54.
doi: 10.1051/m2an:2002002. |
[7] |
G. Barles and E. R. Jakobsen, Error bounds for monotone approximation schemes for Hamilton-Jacobi-Bellman equations, SIAM J. Numer. Anal., 43 (2005), 540-558.
doi: 10.1137/S003614290343815X. |
[8] |
G. Barles and E. R. Jakobsen, Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations, Mathematics of Computations, 76 (2007), 1861-1893.
doi: 10.1090/S0025-5718-07-02000-5. |
[9] |
I. H. Biswas, E. R. Jakobsen and K. H. Karlsen, Difference quadrature schemes for nonlinear degenerate parabolic integro-pde, SIAM J. Numer. Anal., 48 (2010), 1110-1135.
doi: 10.1137/090761501. |
[10] |
I. H. Biswas, E. R. Jakobsen and K. H. Karlsen, Viscosity solutions for a system of integro-pdes and connections to optimal switching and control of jump-diffusion processes, Applied mathematics and optimization, 62 (2010), 47-80.
doi: 10.1007/s00245-009-9095-8. |
[11] |
O. Bokanowski, N. Forcadel and H. Zidani, Reachability and minimal times for state constrained nonlinear problems without any controllability assumption, SIAM J. Control and Optimization. Doi: 10.1137/090762075, 48 (2010), 4292-4316.
doi: 10.1137/090762075. |
[12] |
O. Bokanowski, A. Picarelli and H. Zidani, Dynamic programming and error estimates for stochastic control problems with maximum cost, Applied Math. and Optimization, 71 (2015), 125-163.
doi: 10.1007/s00245-014-9255-3. |
[13] |
J. Bonnans, S. Maroso and H. Zidani, Error bounds for stochastic differential games: The adverse stopping case, IMA, J. Numer. Anal., 26 (2006), 188-212.
doi: 10.1093/imanum/dri034. |
[14] |
J. Bonnans, S. Maroso and H. Zidani, Error estimates for a stochastic impulse control problem, Applied. Math. and Optimisation, 55 (2007), 327-357.
doi: 10.1007/s00245-006-0865-2. |
[15] |
B. Bouchard, R. Elie and N. Touzi, Stochastic target problems with controlled loss, SIAM, 48 (2008), 3123-3150.
doi: 10.1137/08073593X. |
[16] |
A. Briani, F. Camilli and H. Zidani, Approximation schemes for monotone systems of nonlinear second order partial differential equations: convergence result and error estimate, Differential Equations and Applications, 4 (2012), 297-317.
doi: 10.7153/dea-04-18. |
[17] |
L. Caffarelli and P. E. Souganidis, A rate of convergence for monotone finite difference approximations to fully nonlinear uniformly elliptic pde, Comm. Pure Appl. Math., 61 (2008), 1-17.
doi: 10.1002/cpa.20208. |
[18] |
F. Camilli and E. R. Jakobsen, A finite element like scheme for integro-partial differential Hamilton-Jacobi-Bellman equations, SIAM J. Numer. Anal., 47 (2009), 2407-2431.
doi: 10.1137/080723144. |
[19] |
F. Camilli and M. Falcone, An approximation scheme for the optimal control of diffusion processes, RAIRO Modél. Math. Anal. Numér., 29 (1995), 97-122. |
[20] |
F. Da Lio and O. Ley, Uniqueness results for convex Hamilton-Jacobi equations under $p>1$ growth conditions on data, Applied Math. and Optimization, 63 (2011), 309-339. |
[21] |
K. Debrabant and E. R. Jakobsen, Semi-Lagrangian schemes for linear and fully non-linear diffusion equations, Mathematics of Computations, 82 (2013), 1433-1462.
doi: 10.1090/S0025-5718-2012-02632-9. |
[22] |
F. Delarue and S. Menozzi, Density estimates for a random noise propagating through a chain of differential equations, J. Funct. Anal., 259 (2010), 1577-1630.
doi: 10.1016/j.jfa.2010.05.002. |
[23] |
W. H. Fleming and M. H. Soner, Controlled Markov Processes and Viscosity Solutions, vol. 25 of Stochastic Modelling and Applied Probability, 2nd edition, Springer, New York, 2006. |
[24] |
H. Föllmer and P. Leukert, Quantile hedging, SIAM, 3 (1999), 251-273.
doi: 10.1007/s007800050062. |
[25] |
D. Goreac and O.-S. Serea, Mayer and optimal stopping stochastic control problems with discontinuous cost, J. Math. Anal. Appl., 380 (2011), 327-342.
doi: 10.1016/j.jmaa.2011.02.039. |
[26] |
N. V. Krylov, Mean value theorems for stochastic integrals, Ann. Probab., 29 (2001), 385-410.
doi: 10.1214/aop/1008956335. |
[27] |
N. Krylov, On the rate of convergence of finite difference approximation for Bellman's equation, St. Petersburg Math. J., 9 (1998), 639-650. |
[28] |
N. Krylov, On the rate of convergence of finite-difference approximations for Bellman's equations with variable coefficients, Probability Theory and Related Fields, 117 (2000), 1-16.
doi: 10.1007/s004400050264. |
[29] |
N. Krylov, On the rate of convergence for finite-difference approximations for bellman equations with lipschitz coefficients, Applied Mathematics and Optimization, 52 (2005), 365-399.
doi: 10.1007/s00245-005-0832-3. |
[30] |
H. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, vol. 24 of Applications of mathematics, Springer, New York, 2001, Second edition.
doi: 10.1007/978-1-4613-0007-6. |
[31] |
I. Mitchell, A. Bayen and C. Tomlin, A time-dependent Hamiliton-Jacobi formulation of reachable sets for continuous dynamic games, IEEE Transactions on automatic control, 50 (2005), 947-957.
doi: 10.1109/TAC.2005.851439. |
[32] |
R. Munos and H. Zidani, Consistency of a simple multidimensional scheme for hjb equations, C. R. Acad. Sci. Paris, Ser. I, 340 (2005), 499-502.
doi: 10.1016/j.crma.2005.02.001. |
[33] |
S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49.
doi: 10.1016/0021-9991(88)90002-2. |
[34] |
R. Rubinstein and D. Kroese, Simulation and the Monte Carlo Method, Wiley, 2008, Second edition. |
[35] |
J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Stochastic Modelling and Applied Probability, Springer, Paris, 1999.
doi: 10.1007/978-1-4612-1466-3. |
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