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Topological Methods in Nonlinear Analysis

A dual representation result for value functions in stochastic control of infinite dimensional groups
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A dual representation result for value functions in stochastic control of infinite dimensional groups

Authors

  • Viorel Barbu
  • Mihai Sîrbu

Keywords

value function, Legendre transform, duality

Abstract

We study here the problem of dual representation of the value functions associated to linear-convex stochastic control problems in infinite dimensional Hilbert spaces. Since the dual state runs backwards in time, it turns out that the dual representation has the meaning of a classical (Markov) control problem only if the primal linear state equation is driven by the generator of a group. In the general case, a dual representation of the value function still holds, but such a representation cannot be reduced to solving a dual Hamilton-Jacobi-Bellman equation.

References

V. Barbu, Mathematical methods in optimization of differential systems, Mathematics and its Applications, vol. 310, Kluwer Academic Publishers, 1994.

V. Barbu and T. Precupanu, Convexity and optimization in banach spaces, Sijthoff and Noordhoff, 1878.

J.M. Bismut, Conjugate convex functions in optimal stochastic control, J. Math. Anal. Appl. 44 (1973), 384–404.

G. Da Prato and J Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.

I. Karatzas, J.P. Lehoczky, S.E. Shreve and G.L. Xu, Martingale and duality methods for utility maximization in an incomplete market, SIAM J. Control and Optim. 29 (1999), 702–730.

D. Kramkov and W. Schachermayer, The asymptotic elasticity of utility functions and optimal investment in incomplete markets, Ann. Appl. Probab. 9 (1999), 904–950.

M. Sı̂rbu and G. Tessitore, Null controllability of an infinite dimensional sde with stateand-control dependent noise, Systems and Control Letters 44 (2001), 385–394.

G. Tessitore, Existence, uniqueness and space regularity of the adapted solutions of a backward spde, Stoch. Anal. Appl. 14 (1996), 461–486.

J. Yong and X.Y. Zhou, Stochastic controls: Hamiltonian systems and HJB equations, Stochastic Modeling and Applied Probability, vol. 43, Springer–Verlag, 1999.

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Published

2019-11-09

How to Cite

1.
BARBU, Viorel and SÎRBU, Mihai. A dual representation result for value functions in stochastic control of infinite dimensional groups. Topological Methods in Nonlinear Analysis. Online. 9 November 2019. Vol. 54, no. 2B, pp. 907 - 916. [Accessed 7 July 2025].
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Issue

Vol 54, No 2B (December 2019)

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Articles

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