Applied Mathematics & Optimization

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

Stochastic control with rough paths

  • Joscha Diehl
  • Peter K. FrizEmail author
  • Paul Gassiat


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).


Stochastic control Duality Rough paths 

Mathematics Subject Classification

Primary 60H99 



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.


  1. 1.
    Bardi, M., Capuzzo, I.: Dolcetta: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhauser, Boston (1997)CrossRefzbMATHGoogle Scholar
  2. 2.
    Barles, G., Jakobsen, E.R.: On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations. Math. Model. Numer. Anal. 36(1), 33–54 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brown, D.B., Smith, J.E., Sun, P.: Information relaxations and duality in stochastic dynamic programs. Oper. Res. 58(4–Part–1), 785–801 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Buckdahn, R., Ma, J.: Pathwise stochastic control problems and stochastic HJB equations. SIAM J. Control Optim. 45(6), 2224–2256 (2007)Google Scholar
  5. 5.
    Crisan, D., Diehl, J., Friz, P., Oberhauser, H.: Robust filtering: correlated noise and multidimensional observation. Ann. Appl. Prob. 23(5), 2139–2160 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Caruana, M., Friz, P., Oberhauser, H.: A (rough) pathwise approach to a class of nonlinear SPDEs Annales de l’Institut Henri Poincaré / Analyse non linéaire. 28, 27–46 (2011)Google Scholar
  7. 7.
    Coutin, L., Friz, P., Victoir, N.: Good rough path sequences and applications to anticipating stochastic calculus. Ann. Probab. 35(3), 1172–1193 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Davis, M.H.A., Burstein, G.: A deterministic approach to stochastic optimal control with application to anticipative optimal control. Stoch. Stoch. Rep. 40, 203–256 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Diehl, J.: Topics in Stochastic Differential Equations and Rough Path Theory. TU Berlin PhD thesisGoogle Scholar
  10. 10.
    Diehl, J., Friz, P., Oberhauser, H.: Regularity theory for rough partial differential equations and parabolic comparison revisited. In: Crisan et al. (eds.) Springer Proceedings in Mathematics & Statistics, vol. 100, pp. 203–238. Terry Lyons Festschrift, Stochastic Analysis and Applications (2014)Google Scholar
  11. 11.
    Doss, H.: Liens entre équations diffé rentielles stochastiques et ordinaires. Annales de l’institut Henri Poincaré (B) Probabilités et Statistiques, 13(2). Gauthier-Villars (1977)Google Scholar
  12. 12.
    Duncan, T.E., Pasik-Duncan, Bozenna: Linear-quadratic fractional Gaussian control. SIAM J. Control Optim. 51(6), 4504–4519 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions. Springer, New York (1993)zbMATHGoogle Scholar
  14. 14.
    Friz, P., Hairer, M.: A Course on Rough Paths: With an Introduction to Regularity Structures. Springer, New York (2014)CrossRefzbMATHGoogle Scholar
  15. 15.
    Friz, P., Victoir, N.: Multidimensional Stochastic Processes as Rough Paths. Theory and Applications. Cambridge University Press, Cambridge (2010)CrossRefzbMATHGoogle Scholar
  16. 16.
    Gubinelli, M.: Controlling rough paths. J. Funct. Anal. 216, 86–140 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Haugh, M.B., Kogan, L.: Pricing American options: a duality approach. Oper. Res. 52(2), 258–270 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Davis, M.H.A., I. Karatzas. A deterministic approach to optimal stopping. In: Kelly, F.P. (ed.) Probability, Statistics and Optimisation, pp. 455–466. Wiley, Chichester (1994)Google Scholar
  19. 19.
    Krylov, N.: Controlled Diffusion Processes. Springer, New York (1980)CrossRefzbMATHGoogle Scholar
  20. 20.
    Krylov, N.V.: On the rate of convergence of finite-difference approximations for Bellman’s equations with variable coefficients. Probab. Theory Relat. Fields 117(1), 1–16 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Lions, P.L., Souganidis, P.E.: Fully nonlinear stochastic pde with semilinear stochastic dependence. Comptes Rendus de l’Acad mie des Sciences-Series I-Mathematics 331(8), 617–624 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lions, P.-L., Souganidis, P.E.: Fully nonlinear stochastic partial differential equations: non-smooth equations and applications. C.R. Acad. Sci. Paris Ser. I 327, 735–741 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lyons, T.J., et al.: Differential equations driven by rough paths: École d’été de probabilités de Saint-Flour XXXIV-2004. Springer (2007)Google Scholar
  24. 24.
    Lyons, T.J.: Differential equations driven by rough signals. Revista Matemática Iberoamericana 14(2), 215–310 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lyons, T., Qian, Z.: System control and rough paths. Oxford University Press, Oxford (2003)zbMATHGoogle Scholar
  26. 26.
    Mazliak, L., Nourdin, I.: Optimal control for rough differential equations. Stoch. Dyn. 08, 23 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Nualart, D., Pardoux, E.: Stochastic calculus with anticipating integrands. Probab. Theory Relat. Fields 78, 535–581 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Ocone, D., Étienne, P.: A generalized Itô-Ventzell formula. Application to a class of anticipating stochastic differential equations. Annales de l’institut Henri Poincaré (B) Probabilités et Statistiques 25(1), Gauthier-Villars (1989)Google Scholar
  29. 29.
    Perkowski, N., David J.P.: Pathwise stochastic integrals for model free finance. arXiv preprint arXiv:1311.6187 (2013)
  30. 30.
    Rogers, L.C.G.: Pathwise stochastic optimal control. SIAM J. Control Optim. 46(3), 1116–1132 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Rogers, L.C.G.: Monte Carlo valuation of American options. Math. Financ. 12, 271–286 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Sussmann, H.J.: On the gap between deterministic and stochastic ordinary differential equations. Ann. Probab. 6(1), 19–41 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Wets, R.J.B.: On the relation between stochastic and deterministic optimization. Control theory, numerical methods and computer systems modelling. Lecture Notes in Economics and Mathematical Systems, vol. 107, pp. 350–361. Springer (1975)Google Scholar
  34. 34.
    Yong, J., Zhou, X.Y.: Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, New Year (1999)CrossRefzbMATHGoogle Scholar
  35. 35.
    Yosida, K.: Functional Analysis, 6th edn. Springer, New York (1980)zbMATHGoogle Scholar

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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