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1

A direct approach to linear-quadratic stochastic control

Duncan T. E., Pasik-Duncan B.

Opuscula Mathematica

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2017

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Vol. 37, no. 6

821--827

EN

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.

2

Computational Methods for Stochastic Differential Equations and Stochastic Partial Differential Equations Involving Standard Brownian and Fractional Brownian Motion

Shea J., Zachariou I., Pasik-Duncan B.

Challenges of Modern Technology

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2011

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Vol. 2, no. 2

3--12

EN

As more applied science researchers are attempting to use Stochastic Differential Equations (SDEs) as well as Stochastic Partial Differential Equations (SPDEs) in their modeling, especially when involving Fractional Brownian Motion (fBM), one common issue appears: an exact solution cannot always be found. For cases involving SPDEs, exact solutions commonly do not exist and approximation schemes for their solution are typically still in development. Therefore, in this paper, we test various Numerical methods in solving SDEs and SPDEs with standard BM that have non-linear coeffi cients. In addition we extend our results to problems with fBM.

3

Risk sensitive adaptive control of discrete time Markov processes

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

Probability and Mathematical Statistics

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2001

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Vol. 21, Fasc. 2

493--512

EN

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.