A stochastic representation for the level set equations

Communications in Partial Differential Equations

Volumn 27, Issue 9-10, 2002, Pages 2031-2053

  Soner, H Mete ^a   Touzi, Nizar ^b  

^a KOÇ UNIVERSITY   (Turkey)

^b CREST   (France)

Author keywords

Codimension k mean curvature flow; Geometric flows; Inverse mean curvature flow; Stochastic target problem

Indexed keywords

EID: 0036908999     PISSN: 03605302     EISSN: None     Source Type: Journal    
DOI: 10.1081/PDE-120016135     Document Type: Article

References (24)

1. Ambrosio, L.; Soner, H.M. Level Set Approach to Mean Curvature Flow in Arbitrary Codimension. Journal of Differential Geometry 1996, 43, 693-737.
2. Barles, G.; Soner, H.M.; Souganidis, P.E. Front Propagation and Phase Field Theory. SIAM Journal on Control and Optimization, issue Dedicated to W.H. Fleming 1993, 439-469.
3. Bouchard, B. Stochastic Target Problem with Jump-Diffusion Process. Stochastic Processes and Their Applications 2000, forthcoming.
4. Buckdahn, R.; Cardalliaguet, P.; Quincompoix, M. A Representation Formula for the Mean Curvature Motion. SIAM J. Math. Analysis, 2002, 33/4 827-846.
5. Brakke, K. The Motion of a Set By Its Mean Curvature; Princeton University Press: Princeton, NJ, 1978.
6. Chen, Y.-G.; Giga, Y.; Goto, S. Uniqueness and Existence of Viscosity Solutions of Generalized Mean Curvature Flow Equations. Journal of Differential Geometry 1991, 33, 749-786.
7. Crandall, M.; Ishii, H.; Lions, P.-L. User's Guide to Viscosity Solutions of Second Order Partial Differential Equations. Bulletin of AMS 1992, 27, 1-67.
8. Cvitanić, J.; Karatzas, I.; Soner, H.M. Backward SDE's with Constraints on the Gains Process. Annals of Probability 1998, 26(4), 1522-1551.
9. Evans, L.C.; Spruck, J. Motion of Level Sets By Mean Curvature. Journal of Differential Geometry 1991, 33, 635-681.
10. ElKaroui, N.; Nguyen, D.H.; Jeanblanc, M. Compactification Methods in the Control of Degenerate Diffusions: Existence of an Optimal Control. Stochastics 1986, 20, 169-219.
11. Fleming, W.H.; Soner, H.M. Controlled Markov Processes and Viscosity Solutions; Springer-Verlag: New York, Heidelberg, Berlin, 1993.
12. Ilmanen, Tom. Generalized Flow of Sets By Mean Curvature on a Manifold. Indiana Univ. Math. J. 1992, 41(3), 671-705.
13. Karatzas, I.; Shreve, S. Methods of Mathematical Finance; Springer-Verlag: New York, Heidelberg, Berlin, 1998.
14. Haussmann, U.G. Existence of Optimal Markovian Controls for Degenerate Diffusions. Proceedings of The Third Bad-Honnef Conference, Lecture Notes in Control and Information Sciences, Springer-Verlag, 1985.
15. Huisken, G.; Ilmanen, T. The Riemannian Penrose Inequality. Int. Math. Research Notes 1997, 20, 1045-1058.
16. Ma, J.; Yong, J. Forward-Backward Stochastic Differential Equations and Their Applications, Lecture Notes in Mathematics 1702, Springer-Verlag, New York, Heidelberg, Berlin, 1999.
17. Okta, T.; Jasnow, D.; Kawasaki, K. Universal Scalings in the Motion of a Random Interface. Phys. Rev. Lett. 1982, 49, 1223-1226.
18. Osher, S.; Sethian, J. Fronts Propogating with Curvature Dependent Speed. J. Comp. Phys. 1988, 79, 12-49.
19. Soner, H.M. Motion of a Set by the Curvature of Its Boundary. Journal of Differential Equations 1993, 101, 313-372.
20. Soner, H.M.; Touzi, N. Super-Replication Under Gamma Constraints. SIAM J. Control and Opt. 2000, 39, 73-96.
21. Soner, H.M.; Touzi, N. Stochastic Target Problems, Dynamic Programming, and Viscosity Solutions. SIAM J. Control and Opt. 2002, 41, 404-444.
22. Soner, H.M.; Touzi, N. Dynamic Programming for Stochastic Target Problems and Geometric Flows. J. European Math. Soc. 2002, 8, 201-236.
23. Soner, H.M.; Touzi, N. Stochastic Representation of Mean Curvature Type Geometric Flows. Annals of Probability, 2002, forthcoming.
24. Touzi, N. Direct Characterization of The Value of Super-Replication Under Stochastic Volatility and Portfolio Constraints. Stochastic Processes and Their Applications 2000, 88, 305-328.