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Applied Mathematics & Optimization

, Volume 75, Issue 2, pp 285–315 | Cite as

Stochastic control with rough paths

  • Joscha Diehl
  • Peter K. Friz
  • Paul Gassiat
Article

Abstract

We study a class of controlled differential equations driven by rough paths (or rough path realizations of Brownian motion) in the sense of Lyons. It is shown that the value function satisfies a HJB type equation; we also establish a form of the Pontryagin maximum principle. Deterministic problems of this type arise in the duality theory for controlled diffusion processes and typically involve anticipating stochastic analysis. We make the link to old work of Davis and Burstein (Stoch Stoch Rep 40:203–256, 1992) and then prove a continuous-time generalization of Roger’s duality formula [SIAM J Control Optim 46:1116–1132, 2007]. The generic case of controlled volatility is seen to give trivial duality bounds, and explains the focus in Burstein–Davis’ (and this) work on controlled drift. Our study of controlled rough differential equations also relates to work of Mazliak and Nourdin (Stoch Dyn 08:23, 2008).

Keywords

Stochastic control Duality Rough paths 

Mathematics Subject Classification

Primary 60H99 

Notes

Acknowledgments

This work was commenced while all authors were affiliated to TU Berlin. The work of JD was supported by DFG project SPP1324 and DAAD/Marie Curie programme P.R.I.M.E. PKF and PG have received partial funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement nr. 258237 “RPT”. PKF acknowledges support from DFG project FR 2943/2.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.University of California San DiegoLa JollaUSA
  2. 2.TU & WIAS BerlinBerlinGermany
  3. 3.CEREMADE, Université Paris-Dauphine, PSL Research UniversityParisFrance

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