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The Dirichlet problem for semilinear second-order degenerate elliptic equations and applications to stochastic exit time control problems. (English) Zbl 0826.35038

The authors consider the Dirichlet problem for second order semilinear degenerate partial differential equations \[ - {1 \over 2} \sum_{i,j} a_{ij} {\partial^2u \over \partial x_i \partial x_j} + H(x,u,Du) \text{ in } \Omega, \] and the connections of these problems with stochastic exit time control problems.
At first the authors provide a wide historical review of research. Further they recall the definition of viscosity solutions. Then they describe some basic properties of these solutions in the case of the Dirichlet problem for semilinear degenerate elliptic equations. Section II is denoted to state precisely the strong comparison result and to present its consequences for the Dirichlet problem. The existence and uniqueness of continuous of this problem through on approximation by an elliptic regularisation are proved. Next they consider the stochastic exit time control problems. The cost function \[ J \bigl( x, (\alpha_s)_s \bigr) = E_x \left[ \int^\tau_0 f(\alpha_1) \exp (- \lambda t) dt + \varphi (X_\tau) \exp (-\lambda \tau) \right], \] where \(\tau\) is the first exit time of the trajectory \((X_t)_t\) from the set \(\Omega\) and set \(u(x) = \displaystyle\inf_{(\alpha_s)_s} J(x, (\alpha_s)_s)\). In addition, state-constraint problems are considered. For this type of problems, the dynamics is still given by the stochastic differential equation, but the function \(u\) is defined by \(u(x) = \inf E_x [\int^\infty_0 f(X_t, \alpha_t) \exp (- \lambda t) dt]\). In both cases \(u(x)\) is continuous and the unique solution of the Hamilton-Jacobi-Bellman problem.

MSC:

35J70 Degenerate elliptic equations
35J60 Nonlinear elliptic equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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[1] Barles G., Fully nonlinear Neumann type boundary conditions for second-order elliptic and parabolic equations. 106 pp 90– (1993) · Zbl 0786.35051
[2] Barles G., Fully nonlinear Neumann type boundary conditions for first-order Hamilton-Jacobi Equations. 16 pp 143– (1991) · Zbl 0736.35023
[3] Barles G., Discontinuous solutions of deterministic optimal stopping time problems. 21 pp 557– (1987) · Zbl 0629.49017
[4] Barles G., Exit time problems in optimal control and vanishing viscosity method. 26 pp 1133– (1988) · Zbl 0674.49027
[5] Barles G., Comparison principle for Dirichlet type Hamilton-Jacobi Equations and singular perturbations of degenerated elliptic equations. 21 pp 21– (1990) · Zbl 0691.49028
[6] Barles G., Convergence of approximation schemes for fully nonlinear second order equations. 4 pp 271– (1991) · Zbl 0729.65077
[7] Bensoussan, A. 1982. ”STOCHASTIC CONTROL BY FUNCTIONAL ANALYSIS METHODS.”. North Holland: Amsterdam. · Zbl 0474.93002
[8] Bensoussan, A. and Lions, J. L. 1978. ”APPLICATIONS DES INÉQUATIONS VARIA-TIONNELLES EN CONTROLE STOCHASTIQUE”. Dunod
[9] Bensoussan, A. and Lions, J.L. 1972. ”INEQUATIONS VARIATIONNELLES EN CONTROLE STOCHASTIQUE ET EN CONTROLE IMPULSIONNEL.”. Dunod, Paris
[10] Borkar, V.S. 1989. ”OPTIMAL CONTROL OF DIFFUSION PROCESSES.”. Vol. 203, Harlow, UK: Longman Sci. and Tech. Pitman Research Notes · Zbl 0669.93065
[11] Capuzzo-Dolcetta I., Viscosity solutions of Hamilton-Jacobi Equations and state-constraints problems. 318 pp 643– (1990) · Zbl 0702.49019
[12] Crandall M.G, Some properties of viscosity solutions of Hamilton- Jacobi Equations 282 pp 487– (1984) · Zbl 0543.35011
[13] Crandall M.G, User’s guide to viscosity solutions of second order Partial differential equations. 27 pp 1– (1992) · Zbl 0755.35015
[14] Crandall M.G, Viscosity solutions of Hamilton-Jacobi Equations 277 pp 1– (1983)
[15] Dupuis P., On oblique derivative problems for fully nonlinear second-order equations on nonsmooth domains. 12 pp 1123– (1991) · Zbl 0741.35019
[16] Dupuis P., On oblique derivative problems for fully nonlinear second-order elliptic PDE’s on domains with corners. 20 pp 135– (1991) · Zbl 0741.35019
[17] El Karoui, N. 1981. ”LES ASPECTS PROBABILISTES DU CONTROLE STOCHASTIQUE.”. Vol. 876, New-York: Springer-Verlag. Springer Lecture Notes in Math. N{\({}^o\)}
[18] Fichera G., Sulle equazioni differenziali lineari ellitticoparaboliche del secondo ordine. 5 pp 1– (1956) · Zbl 0075.28102
[19] Fichera G., On a unified theory for boundary value problems for elliptic-parabolic equations of second order. pp 97– (1960) · Zbl 0122.33504
[20] Fleming, W.H and Soner, H.M. 1993. ”CONTROLLED MARKOV PROCESSES AND VISCOSITY SOLUTIONS.”. NewYork: Springer-Verlag. Applications of Mathematics · Zbl 0773.60070
[21] Fleming, W.H and Rishel, R.W. 1975. ”DETERMINISTIC AND STOCHASTIC OPTIMAL CONTROL.”. Berlin: Springer. · Zbl 0323.49001
[22] Freidlin, M.I. 1985. ”FUNCTIONAL INTEGRATION AND PARTIAL DIFFERENTIAL EQUATIONS.”. Vol. 109, Princeton University Press. Annals of Math. Studies N{\({}^o\)} · Zbl 0568.60057
[23] Freidlin M.I, On factorisation of a nonnegative definite matrix. 13 pp 375– (1968) · Zbl 0164.46701
[24] Gilbarg, D. and Trudinger, N.S. 1983. ”ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS OF SECOND-ORDER.”. New-York: Springer. · Zbl 0562.35001
[25] Hërmander L., Hypoelliptic second-order differential equations. 119 pp 147– (1967) · Zbl 0156.10701
[26] Ishii H., Perron’ s method for Hamilton-Jacobi Equations. 55 pp 369– (1987) · Zbl 0697.35030
[27] Ishii H., A boundary value problem of the Dirichlet type for Hamilton-Jacobi Equations. 16 pp 105– (1987)
[28] Ishii H., On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDE’s. 42 pp 14– (1989) · Zbl 0645.35025
[29] Ishii H., Fully nonlinear oblique derivative problems for nonlinear second-order elliptic PDE’s. 62 pp 663– (1991) · Zbl 0733.35020
[30] Ishii H., Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. 83 pp 26– (1990) · Zbl 0708.35031
[31] Ivanov A.V, Quasilinear degenerate and non uniformly elliptic and parabolic equations of second order. 198 (1990)
[32] Jensen R., The maximum principe for viscosity solutions of fully nonlinear second-order partial differential equations 101 pp 1– (1988) · Zbl 0708.35019
[33] Jensen R., Uniqueness criteria, for viscosity solutions of fully nonlinear elliptic partial differential equations. 38 pp 629– (1989) · Zbl 0838.35037
[34] Jensen R., A uniqueness result for viscosity solutions of second-order fully nonlinear partial differential equations. 102 pp 975– (1988) · Zbl 0662.35048
[35] Katsoulakis M., Viscosity solutions of 2nd order fully nonlinear elliptic equations with state constraints. · Zbl 0819.35057
[36] Keldysh M.V, On some cases of degenerate elliptic equations. 77 pp 181– (1951)
[37] Kohn J.J, Degenerate Elliptic-Parabolic Equations of Second-Order. pp 797– (1967) · Zbl 0153.14503
[38] Krylov, N.V. 1980. ”CONTROLLED DIFFUSION PROCESSES.”. Springer-Verlag. · Zbl 0459.93002
[39] Ladyzhenskaya, O.A and Ural’tseva, N.N. 1968. ”LINEAR AND QUASILINEAR ELLIPTIC EQUATIONS.”. Academic Press. Traduction franÇaise Dunod
[40] Lasry J.M., Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints. 283 pp 583– (1989) · Zbl 0688.49026
[41] Lions, P.L. 1982. ”GENERALEZED SOLUTIONS OF HAMILTON-JACOBI EQUATIONS”. Pitman.
[42] Lions P.L., Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, Part I: The dynamic programming principle and applcations, 8 pp 1101– (1983)
[43] DOI: 10.1080/03605308308820301 · Zbl 0716.49023 · doi:10.1080/03605308308820301
[44] Lions P.L., Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, Part III, (1985)
[45] Lions P.L, Neumann type boundary conditions for Hamilton-Jacobi Equations. 52 pp 793– (1985) · Zbl 0599.35025
[46] Oleinik O.A, Alcuni risultati sulle Equazioni iineari e quasi lineari Ellitico-Paraboliche a derivaie parziali del second ordine. 40 pp 775– (1966)
[47] Oleinik O.A, A problem of Fichera. 5 pp 1129– (1964)
[48] Oleinik O.A, A boundary value problem for linear elliptic parabolic equations. (1965)
[49] Oleinik O.A, On tie smoothness of solutions of degenerate elliptic and parabolic equations. 163 pp 577– (1965)
[50] Oleinik O.A, On linear second order equations with non negative characteristic form. 69 pp 111– (1966)
[51] Oleinik, O.A and Radkevic, E.V. 1973. ”SECOND-ORDER EQUATIONS WITH NON NEGATIVE CHARACTERISTIC FORM.”. RI: American Mathematical Society. Providence
[52] Phillips R.S, Elliptic Parabolic equations of second-order. 17 pp 891– (1968) · Zbl 0163.34402
[53] Soner M.H, Optimal control problems with state-space constraints. 24 pp 552– (1986) · Zbl 0597.49023
[54] Sontag, E.D. 1990. ”MATHEMATICAL CONTROL THEORY.”. New-York: Springer Verlag. · Zbl 0703.93001
[55] Stroock D.W, On degenerate elliptic-parabolic operators of second order and their associated diffusions. pp 651– (1972) · Zbl 0344.35041
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