Abstract
Solutions of stochastic Volterra (integral) equations are not Markov processes, and therefore, classical methods, such as dynamic programming, cannot be used to study optimal control problems for such equations. However, we show that using Malliavin calculus, it is possible to formulate modified functional types of maximum principle suitable for such systems. This principle also applies to situations where the controller has only partial information available to base her decisions upon. We present both a Mangasarian sufficient condition and a Pontryagin-type maximum principle of this type, and then, we use the results to study some specific examples. In particular, we solve an optimal portfolio problem in a financial market model with memory.
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Acknowledgments
We are grateful to Nils Christian Framstad for helpful comments. The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant Agreement No. [228087]. This research was carried out with support of CAS—Centre for Advanced Study, at the Norwegian Academy of Science and Letters, within the research program SEFE.
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Agram, N., Øksendal, B. Malliavin Calculus and Optimal Control of Stochastic Volterra Equations. J Optim Theory Appl 167, 1070–1094 (2015). https://doi.org/10.1007/s10957-015-0753-5
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DOI: https://doi.org/10.1007/s10957-015-0753-5