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Error estimates for second order Hamilton-Jacobi-Bellman equations. Approximation of probabilistic reachable sets

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  • This work deals with numerical approximations of unbounded and discontinuous value functions associated to some stochastic control problems. We derive error estimates for monotone schemes based on a Semi-Lagrangian method (or more generally in the form of a Markov chain approximation). A motivation of this study consists in approximating chance-constrained reachability sets. The latters will be characterized as level sets of a discontinuous value function associated to an adequate stochastic control problem. A precise analysis of the level-set approach is carried out and some numerical simulations are given to illustrate the approach.
    Mathematics Subject Classification: Primary: 65M15, 93E20; Secondary: 49L25.

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  • [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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