Book Review
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MathSciNet review:
1568111
Full text of review:
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Book Information:
Author:
W. H. Fleming and \par H. M. Soner
Title:
Controlled Markov processes and viscosity solutions
Additional book information:
Applications of Mathematics, volume 25, Springer-Verlag, New York, 1993, xv+428 pp., US$49.95. ISBN 0-387-97927-1.
Richard Bellman, Dynamic programming, Princeton University Press, Princeton, N. J., 1957. MR 0090477
M. H. A. Davis, Linear estimation and stochastic control, Chapman and Hall Mathematics Series, Chapman and Hall, London; Halsted Press [John Wiley & Sons, Inc.], New York, 1977. MR 0476099
[3] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mischenko, The mathematical theory of optimal processes, Interscience, New York, 1962.
D. Q. Mayne and E. Polak, First-order strong variation algorithms for optimal control, J. Optim. Theory Appl. 16 (1975), 277–301. MR 373284, DOI 10.1007/BF01262937
Wendell H. Fleming and Raymond W. Rishel, Deterministic and stochastic optimal control, Applications of Mathematics, No. 1, Springer-Verlag, Berlin-New York, 1975. MR 0454768
[6] F. H. Clarke, Optimization and non-smooth analysis, Wiley-lnterscience, New York, 1983.
Michael G. Crandall and Pierre-Louis Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), no. 1, 1–42. MR 690039, DOI 10.1090/S0002-9947-1983-0690039-8
Michael G. Crandall and Pierre-Louis Lions, Condition d’unicité pour les solutions généralisées des équations de Hamilton-Jacobi du premier ordre, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 3, 183–186 (French, with English summary). MR 610314
Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67. MR 1118699, DOI 10.1090/S0273-0979-1992-00266-5
Robert J. Elliott, Viscosity solutions and optimal control, Pitman Research Notes in Mathematics Series, vol. 165, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987. MR 913938
Michael Crandall, Viscosity solutions of partial differential equations, AMS Progress in Mathematics Lecture Series, American Mathematical Society, Providence, RI, 1991. MR 1146680
G. Barles and B. Perthame, Exit time problems in optimal control and vanishing viscosity method, SIAM J. Control Optim. 26 (1988), no. 5, 1133–1148 (English, with French summary). MR 957658, DOI 10.1137/0326063
G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal. 4 (1991), no. 3, 271–283. MR 1115933
Harold J. Kushner and Paul G. Dupuis, Numerical methods for stochastic control problems in continuous time, Applications of Mathematics (New York), vol. 24, Springer-Verlag, New York, 1992. MR 1217486, DOI 10.1007/978-1-4684-0441-8
Mark H. A. Davis, Vassilios G. Panas, and Thaleia Zariphopoulou, European option pricing with transaction costs, SIAM J. Control Optim. 31 (1993), no. 2, 470–493. MR 1205985, DOI 10.1137/0331022
- [1]
- R. Bellman, Dynamic programming, Princeton Univ. Press, Princeton, NJ, 1957. MR 0090477 (19:820d)
- [2]
- M. H. A. Davis, Linear estimation and stochastic control, Chapman & Hall, London, 1977. MR 0476099 (57:15678)
- [3]
- L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mischenko, The mathematical theory of optimal processes, Interscience, New York, 1962.
- [4]
- D. Q. Mayne and E. Polak, First order strong variation algorithms for optimal control, J. Optim. Theory Appl. 16 (1975), 277-301. MR 0373284 (51:9484)
- [5]
- W. H. Fleming and R. W. Rishel, Deterministic and stochastic optimal control, Springer-Verlag, New York, 1975. MR 0454768 (56:13016)
- [6]
- F. H. Clarke, Optimization and non-smooth analysis, Wiley-lnterscience, New York, 1983.
- [7]
- M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1984), 1-42. MR 690039 (85g:35029)
- [8]
- -, Condition d'unicité pour les solutions généralisées des équations de Hamilton-Jacobi du premier ordre, C. R. Acad. Sci. Paris Sér. I. Math. 292 (1981), 183-186. MR 610314 (82c:49020)
- [9]
- M. G. Crandall, H. Ishii, and P. L. Lions, A user's guide to viscosity solutions, Bull. Amer. Math. Soc. (N.S.) 27 (1992), 1-67. MR 1118699 (92j:35050)
- [10]
- R. J. Elliott, Viscosity solutions and optimal control, Pitman Res. Notes in Math., vol. 165, Longman, London, 1987. MR 913938 (89a:49028)
- [11]
- M. Crandall, Viscosity solutions of partial differential equations, videotape, AMS Progress in Mathematics Series, Amer. Math. Soc., Providence, RI, 1991. MR 1146680 (92k:35001)
- [12]
- G. Barles and B. Perthame, Exit time problems in optimal control and vanishing viscosity solutions of Hamilton-Jacobi equations, SIAM J. Control Optim. 26 (1988), 1133-1148. MR 957658 (89i:49021)
- [13]
- G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, J. Asymptotic Anal. 4 (1991), 271-283. MR 1115933 (92d:35137)
- [14]
- H. J. Kushner and P. Dupuis, Numerical methods for stochastic control problems in continuous time, Springer-Verlag, New York, 1992. MR 1217486 (94e:93005)
- [15]
- M. H. A. Davis, V. G. Panas, and T. Zariphopoulou, European option pricing with transaction costs, SIAM J. Control Optim. 31 (1993), 470-493. MR 1205985 (94d:90012)
Review Information:
Reviewer:
M. H. A. Davis
Journal:
Bull. Amer. Math. Soc.
31 (1994), 75-85
DOI:
https://doi.org/10.1090/S0273-0979-1994-00480-X
