Stochastic control problems and spherical functions on symmetric spaces
Authors:
T. E. Duncan and H. Upmeier
Journal:
Trans. Amer. Math. Soc. 347 (1995), 10831130
MSC:
Primary 93E20; Secondary 43A90, 49J45, 53C35
MathSciNet review:
1284453
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Abstract 
References 
Similar Articles 
Additional Information
Abstract: A family of explicitly solvable stochastic control problems is formulated and solved in noncompact symmetric spaces. The symmetric spaces include all of the classical spaces and four of the exceptional spaces. The stochastic control problems are the control of Brownian motion in these symmetric spaces by a drift vector field. For each symmetric space a family of stochastic control problems is formulated by using spherical functions in the cost functionals. These spherical functions are explicitly described and are polynomials in suitable coordinates. A generalization to abstract root systems is given.
 [BK]
H. Braun and M. Koecher, JordanAlgebren, Springer, 1966.
 [D1]
A. Debiard, Système diffèrentiel hypergéométrique et parties radiales des opérateurs invariants des espaces symétriques de type , Lecture Notes in Math., vol. 1296, Springer, 1988, pp. 42124.
 [D2]
T. E. Duncan, Stochastic systems in Riemannian manifolds, J. Optim. Theory Appl. 27 (1979), 3994210.
 [D3]
, Dynamic programming optimality criteria for stochastic systems in Riemannian manifolds, Appl. Math. Optim. 3 (1977), 191208.
 [D4]
, A solvable stochastic control problem in hyperbolic three space, Systems Control Lett. 8 (1987), 435439.
 [D5]
, Some solvable stochastic control problems in noncompact symmetric spaces of rank one, Stochastics Stochastics Rep. 35 (1991), 129142.
 [DU]
T. E. Duncan and H. Upmeier, Stochastic control problems in symmetric cones and spherical functions, Diffusion Processes and Related Problems in Analysis, Vol. 1, Birkhäuser, 1990, pp. 263283.
 [FK]
J. Faraut and A. Korányi, Analysis on symmetric cones (to appear).
 [F1]
R. H. Farrell, Multivariate calculation, SpringerVerlag, New YorkBerlin, 1985.
 [FR]
W. H. Fleming and R. W. Rishel, Deterministic and stochastic optimal control, SpringerVerlag, 1975.
 [H1]
S. Helgason, Groups and geometric analysis, Academic Press, 1984.
 [H2]
B. Hoogenboom, Spherical functions and differential operators in complex Grassmann manifolds, Ark. Mat. 29 (1982), 6985.
 [H3]
L. Hörmander, The analysis of linear partial differential equations, Vol. I, Springer, 1983.
 [IW]
N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, NorthHolland, 1981.
 [J1]
K. D. Johnson, On a ring of invariant polynomials on a Hermitian symmetric space, J. Algebra 67 (1980), 7281.
 [J2]
A. T. James, Calculations of zonal polynomial coefficients by use of the LaplaceBeltrami operator, Ann. Math. Statist. 39 (1968), 17111718.
 [KW]
A. Korányi and J. A. Wolf, Generalized Cayley transformations of bounded symmetric domains, Amer. J. Math. 87 (1965), 899939.
 [L1]
O. Loos, Symmetric spaces, I, II, Benjamin, 1969.
 [L2]
, Jordan triple systems, spaces and bounded symmetric domains, Bull. Amer. Math. Soc. 77 (1971), 558561.
 [L3]
, Bounded symmetric domains and Jordan pairs, Univ. of California, Irvine, 1977.
 [M1]
I. Macdonald, Symmetric functions and Hall polynomials, Clarendon Press, Oxford, 1979.
 [M2]
, Commuting differential operators and zonal spherical functions, Lecture Notes in Math., vol. 1271, Springer, 1987, pp. 189199.
 [M2]
R. J. Muirhead, Aspects of multivariate statistical theory, Wiley, New York, 1982.
 [N1]
E. Neher, Klassifikation der einfachen reellen JordanTripelsysteme, Dissertation, Münster, 1978.
 [S1]
R. P. Stanley, Some combinatorial properties of Jack symmetric functions, Adv. Math. 77 (1989), 76115.
 [T1]
A. Takemura, Zonal polynomials, Inst. Math. Stat. Lecture NotesMonograph Series, 4, 1984.
 [U1]
H. Upmeier, Symmetric Banach manifolds and Jordan algebras, NorthHolland, 1985.
 [U2]
, Jordan algebras and harmonic analysis on symmetric spaces, Amer. J. Math. 108 (1986), 125.
 [U3]
, Jordan algebras in analysis, Operator Theory and Quantum Mechanics, CMBS Regional Conf. Ser. in Math., no. 67, Providence, RI, 1987.
 [Y1]
Z. Yan, Hypergeometric functions in several variables, Thesis, CUNY, 1990.
 [Y2]
K. Yosida, Functional analysis, Springer, 1971.
 [BK]
 H. Braun and M. Koecher, JordanAlgebren, Springer, 1966.
 [D1]
 A. Debiard, Système diffèrentiel hypergéométrique et parties radiales des opérateurs invariants des espaces symétriques de type , Lecture Notes in Math., vol. 1296, Springer, 1988, pp. 42124.
 [D2]
 T. E. Duncan, Stochastic systems in Riemannian manifolds, J. Optim. Theory Appl. 27 (1979), 3994210.
 [D3]
 , Dynamic programming optimality criteria for stochastic systems in Riemannian manifolds, Appl. Math. Optim. 3 (1977), 191208.
 [D4]
 , A solvable stochastic control problem in hyperbolic three space, Systems Control Lett. 8 (1987), 435439.
 [D5]
 , Some solvable stochastic control problems in noncompact symmetric spaces of rank one, Stochastics Stochastics Rep. 35 (1991), 129142.
 [DU]
 T. E. Duncan and H. Upmeier, Stochastic control problems in symmetric cones and spherical functions, Diffusion Processes and Related Problems in Analysis, Vol. 1, Birkhäuser, 1990, pp. 263283.
 [FK]
 J. Faraut and A. Korányi, Analysis on symmetric cones (to appear).
 [F1]
 R. H. Farrell, Multivariate calculation, SpringerVerlag, New YorkBerlin, 1985.
 [FR]
 W. H. Fleming and R. W. Rishel, Deterministic and stochastic optimal control, SpringerVerlag, 1975.
 [H1]
 S. Helgason, Groups and geometric analysis, Academic Press, 1984.
 [H2]
 B. Hoogenboom, Spherical functions and differential operators in complex Grassmann manifolds, Ark. Mat. 29 (1982), 6985.
 [H3]
 L. Hörmander, The analysis of linear partial differential equations, Vol. I, Springer, 1983.
 [IW]
 N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, NorthHolland, 1981.
 [J1]
 K. D. Johnson, On a ring of invariant polynomials on a Hermitian symmetric space, J. Algebra 67 (1980), 7281.
 [J2]
 A. T. James, Calculations of zonal polynomial coefficients by use of the LaplaceBeltrami operator, Ann. Math. Statist. 39 (1968), 17111718.
 [KW]
 A. Korányi and J. A. Wolf, Generalized Cayley transformations of bounded symmetric domains, Amer. J. Math. 87 (1965), 899939.
 [L1]
 O. Loos, Symmetric spaces, I, II, Benjamin, 1969.
 [L2]
 , Jordan triple systems, spaces and bounded symmetric domains, Bull. Amer. Math. Soc. 77 (1971), 558561.
 [L3]
 , Bounded symmetric domains and Jordan pairs, Univ. of California, Irvine, 1977.
 [M1]
 I. Macdonald, Symmetric functions and Hall polynomials, Clarendon Press, Oxford, 1979.
 [M2]
 , Commuting differential operators and zonal spherical functions, Lecture Notes in Math., vol. 1271, Springer, 1987, pp. 189199.
 [M2]
 R. J. Muirhead, Aspects of multivariate statistical theory, Wiley, New York, 1982.
 [N1]
 E. Neher, Klassifikation der einfachen reellen JordanTripelsysteme, Dissertation, Münster, 1978.
 [S1]
 R. P. Stanley, Some combinatorial properties of Jack symmetric functions, Adv. Math. 77 (1989), 76115.
 [T1]
 A. Takemura, Zonal polynomials, Inst. Math. Stat. Lecture NotesMonograph Series, 4, 1984.
 [U1]
 H. Upmeier, Symmetric Banach manifolds and Jordan algebras, NorthHolland, 1985.
 [U2]
 , Jordan algebras and harmonic analysis on symmetric spaces, Amer. J. Math. 108 (1986), 125.
 [U3]
 , Jordan algebras in analysis, Operator Theory and Quantum Mechanics, CMBS Regional Conf. Ser. in Math., no. 67, Providence, RI, 1987.
 [Y1]
 Z. Yan, Hypergeometric functions in several variables, Thesis, CUNY, 1990.
 [Y2]
 K. Yosida, Functional analysis, Springer, 1971.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199512844538
PII:
S 00029947(1995)12844538
Keywords:
Stochastic control,
spherical functions,
controlled diffusions in symmetric spaces,
LaplaceBeltrami operator,
explicitly solvable control problems
Article copyright:
© Copyright 1995
American Mathematical Society
