A quasi-sure approach to the control of non-Markovian stochastic differential equations

Marcel Nutz (Columbia University)


We study stochastic differential equations (SDEs) whose drift and diffusion coefficients are path-dependent and controlled. We construct a value process on the canonical path space, considered simultaneously under a family of singular measures, rather than the usual family of processes indexed by the controls. This value process is characterized by a second order backward SDE, which can be seen as a non-Markovian analogue of the Hamilton-Jacobi Bellman partial differential equation. Moreover, our value process yields a generalization of the $G$-expectation to the context of SDEs.

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Pages: 1-23

Publication Date: March 19, 2012

DOI: 10.1214/EJP.v17-1892


  • Bichteler, Klaus. Stochastic integration and $L^{p}$-theory of semimartingales. Ann. Probab. 9 (1981), no. 1, 49--89. MR0606798
  • Buckdahn, Rainer; Ma, Jin. Stochastic viscosity solutions for nonlinear stochastic partial differential equations. I. Stochastic Process. Appl. 93 (2001), no. 2, 181--204. MR1828772
  • Buckdahn, Rainer; Ma, Jin. Stochastic viscosity solutions for nonlinear stochastic partial differential equations. II. Stochastic Process. Appl. 93 (2001), no. 2, 205--228. MR1831830
  • Buckdahn, Rainer; Ma, Jin. Pathwise stochastic control problems and stochastic HJB equations. SIAM J. Control Optim. 45 (2007), no. 6, 2224--2256 (electronic). MR2285722
  • Cheridito, Patrick; Soner, H. Mete; Touzi, Nizar; Victoir, Nicolas. Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs. Comm. Pure Appl. Math. 60 (2007), no. 7, 1081--1110. MR2319056
  • Dellacherie, Claude; Meyer, Paul-André. Probabilities and potential. B. Theory of martingales. Translated from the French by J. P. Wilson. North-Holland Mathematics Studies, 72. North-Holland Publishing Co., Amsterdam, 1982. xvii+463 pp. ISBN: 0-444-86526-8 MR0745449
  • Denis, Laurent; Hu, Mingshang; Peng, Shige. Function spaces and capacity related to a sublinear expectation: application to $G$-Brownian motion paths. Potential Anal. 34 (2011), no. 2, 139--161. MR2754968
  • Denis, Laurent; Martini, Claude. A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab. 16 (2006), no. 2, 827--852. MR2244434
  • El Karoui, N. Les aspects probabilistes du contrôle stochastique. (French) [The probabilistic aspects of stochastic control] Ninth Saint Flour Probability Summer School—1979 (Saint Flour, 1979), pp. 73--238, Lecture Notes in Math., 876, Springer, Berlin-New York, 1981. MR0637471
  • Elliott, Robert J. Stochastic calculus and applications. Applications of Mathematics (New York), 18. Springer-Verlag, New York, 1982. ix+302 pp. ISBN: 0-387-90763-7 MR0678919
  • Feldman, J.; Smorodinsky, M. Simple examples of non-generating Girsanov processes. Séminaire de Probabilités, XXXI, 247--251, Lecture Notes in Math., 1655, Springer, Berlin, 1997. MR1478733
  • Fleming, Wendell H.; Soner, H. Mete. Controlled Markov processes and viscosity solutions. Second edition. Stochastic Modelling and Applied Probability, 25. Springer, New York, 2006. xviii+429 pp. ISBN: 978-0387-260457; 0-387-26045-5 MR2179357
  • Hu, Ying; Imkeller, Peter; Müller, Matthias. Utility maximization in incomplete markets. Ann. Appl. Probab. 15 (2005), no. 3, 1691--1712. MR2152241
  • Lions, Pierre-Louis; Souganidis, Panagiotis E. Fully nonlinear stochastic partial differential equations. C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 9, 1085--1092. MR1647162
  • Lions, Pierre-Louis; Souganidis, Panagiotis E. Fully nonlinear stochastic partial differential equations: non-smooth equations and applications. C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), no. 8, 735--741. MR1659958
  • Mandelkern, Mark. On the uniform continuity of Tietze extensions. Arch. Math. (Basel) 55 (1990), no. 4, 387--388. MR1076066
  • M. Nutz, The Bellman equation for power utility maximization with semimartingales. Ann. Appl. Probab. 22 (2012), no. 1, 363--406.
  • bysame, Random G-expectations, Preprint arXiv:1009.2168v1 (2010).
  • M. Nutz and H. M. Soner, Superhedging and dynamic risk measures under volatility uncertainty, Preprint arXiv:1011.2958v1 (2010).
  • Pardoux, É.; Peng, S. G. Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 (1990), no. 1, 55--61. MR1037747
  • Peng, Shi Ge. Stochastic Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 30 (1992), no. 2, 284--304. MR1149069
  • Peng, Shige. Filtration consistent nonlinear expectations and evaluations of contingent claims. Acta Math. Appl. Sin. Engl. Ser. 20 (2004), no. 2, 191--214. MR2064000
  • bysame, G-expectation, G-Brownian motion and related stochastic calculus of Itô type, Stochastic Analysis and Applications (Springer, Berlin), Abel Symp., vol. 2, 2007, pp. 541--567.
  • Peng, Shige. Multi-dimensional $G$-Brownian motion and related stochastic calculus under $G$-expectation. Stochastic Process. Appl. 118 (2008), no. 12, 2223--2253. MR2474349
  • bysame, Note on viscosity solution of path-dependent PDE and G-martingales, Preprint arXiv:1106.1144v1 (2011).
  • H. M. Soner, N. Touzi, and J. Zhang, Dual formulation of second order target problems, To appear in Ann. Appl. Probab. (2010).
  • bysame, Wellposedness of second order backward SDEs, To appear in Probab. Theory Related Fields (2010).
  • Soner, H. Mete; Touzi, Nizar; Zhang, Jianfeng. Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16 (2011), no. 67, 1844--1879. MR2842089
  • Stroock, Daniel W.; Varadhan, S. R. Srinivasa. Multidimensional diffusion processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 233. Springer-Verlag, Berlin-New York, 1979. xii+338 pp. ISBN: 3-540-90353-4 MR0532498

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