Abstract
This paper establishes a general analytical framework for classical and impulse stochastic control problems in the presence of model uncertainty. We consider a set of dominated models, which are induced by the measures equivalent to that of a reference model. The state process under the reference model is a multidimensional Markov process with multidimensional Brownian motion, controlled by continuous and impulse control variates. We propose quasi-variational inequalities (QVI) associated with the value function of the control problem and prove a verification theorem for the solution to the QVI. With the relative entropy constraints and piecewise linear intervention penalty, we show that the QVI can be degenerated to the non-robust case and it can be solved via the solution to a free boundary problem. To illustrate the tractability of the proposed framework, we apply it to a linear-quadratic setting, which covers a broad class of problems including robust mean-reverting inventory controls.
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Notes
In Maenhout (2004), robust portfolio choices are found to be simply enlarging the estimate of the volatility.
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Acknowledgements
Chi Seng Pun gratefully acknowledges Ministry of Education (MOE), AcRF Tier 2 grant (Reference No: MOE2017-T2-1-044) for the funding of this research.
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Pun, C.S. Robust classical-impulse stochastic control problems in an infinite horizon. Math Meth Oper Res 96, 291–312 (2022). https://doi.org/10.1007/s00186-022-00795-9
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DOI: https://doi.org/10.1007/s00186-022-00795-9