ISSN:
1573-2878
Keywords:
Stationary stochastic control
;
Bellman equation
;
invariant measures
;
nonanticipative laws
;
dead-zone controllers
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract 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 $$d\xi _t = u_1 (\xi )dt + dw_t , \xi _0 = x, |u| \leqslant 1,$$ (w t) a Wiener process, with all measurable functions on the past of the state process {ξ s ;s≤t} 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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00934934
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