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A direct approach to linear-quadratic stochastic control

Duncan T. E., Pasik-Duncan B.

Opuscula Mathematica

2017

Vol. 37, no. 6

821--827

A direct approach is used to solve some linear-quadratic stochastic control problems for Brownian motion and other noise processes. This direct method does not require solving Hamilton-Jacobi-Bellman partial differential equations or backward stochastic differential equations with a stochastic maximum principle or the use of a dynamic programming principle. The appropriate Riccati equation is obtained as part of the optimization problem. The noise processes can be fairly general including the family of fractional Brownian motions.

Risk sensitive adaptive control of discrete time Markov processes

Duncan T. E., Pasik-Duncan B., Stettner Ł.

Probability and Mathematical Statistics

2001

Vol. 21, Fasc. 2

493--512

Adaptive control of discrete time Markov processes with an infinite horizon risk sensitive cost functional is investigated. The continuity of the optimal risk sensitive cost with respect to a parameter of the transition probability is verified. Two almost optimal adaptive procedures that are based on the large deviations of the cost functional and discretized maximum likelihood estimates are given. To justify the performance of the adaptive procedure with observations of the cost, some large deviations estimates of the empirical distributions of finite sequences of successive states of Markov processes are obtained. A finite family of continuous control functions, where one control function is fixed after a nonrandom time from each of the adaptive procedures, provides an almost optimal adaptive control.