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

Viorel Barbu, Mihai Sîrbu


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.


value function; Legendre transform; duality

Full Text:



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.


  • There are currently no refbacks.

Partnerzy platformy czasopism