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

Electronic Journal of Probability

Volumn 17, Issue , 2012, Pages

  Nutz, Marcel ^a  

^a Columbia University ^*  (United States)

Author keywords

G expectation; Non Markovian SDE; Random G expectation; Risk measure; Second order BSDE; Stochastic optimal control; Volatility uncertainty

Indexed keywords

EID: 84859098453     PISSN: None     EISSN: 10836489     Source Type: Journal    
DOI: 10.1214/EJP.v17-1892     Document Type: Article

References (29)

1. p-theory of semimartingales, Ann. Probab. 9 (1981), no. 1, 49-89. MR-0606798
2. R. Buckdahn and J. Ma, Stochastic viscosity solutions for nonlinear stochastic partial differential equations. I., Stochastic Process. Appl. 93 (2001), no. 2, 181-204. MR-1828772
3. R. Buckdahn and J. Ma, Stochastic viscosity solutions for nonlinear stochastic partial differential equations. II., Stochastic Process. Appl. 93 (2001), no. 2, 205-228. MR-1831830
4. R. Buckdahn and J. Ma, Pathwise stochastic control problems and stochastic HJB equations, SIAM J. Control Optim. 45 (2007), no. 6, 2224-2256. MR-2285722
5. P. Cheridito, H. M. Soner, N. Touzi, and N. Victoir, Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs, Comm. Pure Appl. Math. 60 (2007), no. 7, 1081-1110. MR-2319056
6. C. Dellacherie and P. A. Meyer, Probabilities and potential B, North Holland, Amsterdam, 1982. MR-0745449
7. L. Denis, M. Hu, and S. Peng, Function spaces and capacity related to a sublinear expectation: application to G-Brownian motion paths, Potential Anal. 34 (2011), no. 2, 139-161. MR-2754968
8. L. Denis and C. Martini, A theoretical framework for the pricing of contingent claims in the presence of model uncertainty, Ann. Appl. Probab. 16 (2006), no. 2, 827-852. MR-2244434
9. N. El Karoui, Les aspects probabilistes du contrôle stochastique, Ecole d'été de probabilitées de Saint-Flour (Springer, Berlin), Lecture Notes in Math., vol. 876, 1981, pp. 73-238. MR-0637471
10. R. J. Elliott, Stochastic calculus and applications, Springer, New York, 1982. MR-0678919
11. J. Feldman and M. Smorodinsky, Simple examples of non-generating Girsanov processes, Séminaire de Probabilités XXXI, Lecture Notes in Math., vol. 1655, Springer, Berlin, 1997, pp. 247-251. MR-1478733
12. W. H. Fleming and H. M. Soner, Controlled Markov processes and viscosity solutions, 2nd ed., Springer, New York, 2006. MR-2179357
13. Y. Hu, P. Imkeller, and M. Müller, Utility maximization in incomplete markets, Ann. Appl. Probab. 15 (2005), no. 3, 1691-1712. MR-2152241
14. P.-L. Lions and P. Souganidis, Fully nonlinear stochastic partial differential equations, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 9, 1085-1092. MR-1647162
15. P.-L. Lions and P. Souganidis, 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. MR-1659958
16. M. Mandelkern, On the uniform continuity of Tietze extensions, Arch. Math. 55 (1990), no. 4, 387-388. MR-1076066
17. M. Nutz, The Bellman equation for power utility maximization with semimartingales, Ann. Appl. Probab. 22 (2012), no. 1, 363-406.
18. M. Nutz, Random G-expectations, Preprint arXiv:1009.2168v1 (2010).
19. M. Nutz and H. M. Soner, Superhedging and dynamic risk measures under volatility uncertainty, Preprint arXiv:1011.2958v1 (2010).
20. E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14 (1990), no. 1, 55-61. MR-1037747
21. S. Peng, Stochastic Hamilton-Jacobi-Bellman equations, SIAM J. Control Optim. 30 (1992), no. 2, 284-304. MR-1149069
22. S. Peng, Filtration consistent nonlinear expectations and evaluations of contingent claims, Acta Math. Appl. Sin. Engl. Ser. 20 (2004), no. 2, 191-214. MR-2064000
23. S. Peng, 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.
24. S. Peng, Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation, Stochastic Process. Appl. 118 (2008), no. 12, 2223-2253. MR-2474349
25. S. Peng, Note on viscosity solution of path-dependent PDE and G-martingales, Preprint arXiv:1106.1144v1 (2011).
26. H. M. Soner, N. Touzi, and J. Zhang, Dual formulation of second order target problems, To appear in Ann. Appl. Probab. (2010).
27. S. Peng, Wellposedness of second order backward SDEs, To appear in Probab. Theory Related Fields (2010).
28. S. Peng, Quasi-sure stochastic analysis through aggregation, Electron. J. Probab. 16 (2011), no. 2, 1844-1879. MR-2842089
29. D. Stroock and S. R. S. Varadhan, Multidimensional diffusion processes, Springer, New York, 1979. MR-0532498