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Finite-Dimensional Representations for Controlled Diffusions with Delay

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Abstract

We study stochastic delay differential equations (SDDE) where the coefficients depend on the moving averages of the state process. As a first contribution, we provide sufficient conditions under which the solution of the SDDE and a linear path functional of it admit a finite-dimensional Markovian representation. As a second contribution, we show how approximate finite-dimensional Markovian representations may be constructed when these conditions are not satisfied, and provide an estimate of the error corresponding to these approximations. These results are applied to optimal control and optimal stopping problems for stochastic systems with delay.

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Notes

  1. When the delay appears also in the control variable the infinite-dimensional representation is more involved. We refer to [3, Part II, Chap. 4], where a general theory is developed based on the paper [34].

  2. Nevertheless there are examples where a finite-dimensional Markovian representation can be obtained. We will study this kind of situation in Sect. 4.2, giving sufficient conditions for a finite-dimensional Markovian representation.

  3. For example this process could appear in the cost functional of a control problem.

  4. In the usual language of stochastic integration in infinite-dimension (see [7, 22, 31]), \(\Sigma (\mathbf {x},u)\in H_w\) should be seen as a Hilbert–Schmidt operator from \(\mathbb {R}\) to \(H_w\).

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Acknowledgments

The authors are grateful to two anonymous Referees whose suggestions allowed to improve the first version of the paper. The authors also thank Mauro Rosestolato for very fruitful discussions and suggestions.

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Correspondence to Salvatore Federico.

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Federico, S., Tankov, P. Finite-Dimensional Representations for Controlled Diffusions with Delay. Appl Math Optim 71, 165–194 (2015). https://doi.org/10.1007/s00245-014-9256-2

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