Journal of Optimization Theory and Applications

, Volume 35, Issue 4, pp 611–633 | Cite as

Optimal stationary linear control of the Wiener process

  • V. E. Beneš
  • I. Karatzas
Contributed Papers


In the present paper, we consider the following stochastic control problem: to minimize the average expected total cost
$$J(x,u) = \mathop {\lim \inf }\limits_{T \to \infty } (1/T)E_x^u \int_0^T {\left[ {\phi (\xi _t ) + |u_t (\xi )|} \right]} dt,$$
〈subject to
$$d\xi _t = u_1 (\xi )dt + dw_t , \xi _0 = x, |u| \leqslant 1,$$
(wt) a Wiener process, with all measurable functions on the past of the state process {ξ s ;st} and bounded by unity, admissible as controls. It is proved that, under very mild conditions on the running cost function φ(·), the optimal law is of the form
$$\begin{gathered} u_t^* (\xi ) = - sign\xi _t , |\xi _t | > b, \hfill \\ u_t^* (\xi ) = 0, |\xi _t | > b. \hfill \\ \end{gathered} $$
The cutoff pointb and the performance rate of the optimal lawu* are simultaneously determined in terms of the function φ(·) through a simple system of integrotranscendental equations.

Key Words

Stationary stochastic control Bellman equation invariant measures nonanticipative laws dead-zone controllers 


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Copyright information

© Plenum Publishing Corporation 1981

Authors and Affiliations

  • V. E. Beneš
    • 1
  • I. Karatzas
    • 2
  1. 1.Bell Telephone LaboratoriesMurray Hill
  2. 2.Division of Applied Mathematics and Lefschetz Center for Dynamical SystemsBrown UniversityProvidence

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