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RESEARCH of sONER

RESEARCH INTERESTS

 

Current interests are nonlinear partial differential equations; asymptotic analysis of Ginzburg-Landau type systems, viscosity solutions, and mathematical finance.  List of all research articles can be found in the section called publications.

 

Currently I am involved in three European Networks; one on partial differential equations HYKE, one on front propagation FRONTS, and the other on mathematical finance, AMaMeF.  In the first two, I am part of the team from Crete.  In August 2008, I received an ERC (European Research Council) Advanced Research Grant for an amount of 880,560 Euros.  This was the only successful application from Turkey in all fields of Engineering and Basic Sciences.

 

RESEARCH SUPPORT AND GRANTS

           

Throughout my tenure in the United States I had continuous support from the National Science Foundation and also from the Air Force Office of Research for six years.  I was the Associate Director for a Center for Nonlinear Analysis at Carnegie Mellon during 1995-1998.  Following is the list of grants received or membership to research networks since 1998.

 

 

PAPERS AND PREPRINTS RELATED TO RECENT RESEARCH

 

·         Super-replication and Stochastic target problems:  For more than a decade, Nizar Touzi and I have been involved in developing tools for problems of super-replication in finance.  We identified a general class of optimal control problems and called it stochastic target problems.  We developed the dynamic programming techniques for these problems and for super-replication.  Related papers and preprints are:

 

1.    Cetin-Jarrow-Protter model in a Binomial market, (with S. Gökay), 2008, preprint.

2.    The dynamic programming equation for second order stochastic target problems, (with N. Touzi), 2007, submitted.

3.    Hedging under Gamma constraints by optimal stopping and face-lifting, (with N. Touzi) Mathematical Finance, 17 (1): 59-79 (2007).

4.    Small time path behavior of double stochastic integrals and applications to stochastic control, (with P. Cheridito and N. Touzi) Annals of Applied Probability,  15 (4): 2472-2495 (2005).

5.     The multi-dimensional super-replication problem under gamma constraints,  (with P. Cheridito and N. Touzi)Annales de L'Isntitute Henri Poincare Analyse Nonlineaire,  22 (5): 633-666 (2005).

6.     A stochastic representation for mean curvature type flows, (with N. Touzi) Annals of Probability, 31/3, 1145-1165, (2003).

7.     A stochastic representation for level set equations, (with N. Touzi) Communications in PDE's, 27(9&10), 2031-2053, (2002).

8.     Dynamic programming for stochastic target problems and geometric flows, (with N. Touzi) Journal of European Mathematical Society, 4/3, 201-236, (2002).

9.    Stochastic target problems and dynamic programming, (with N. Touzi) SIAM Journal on Control and Optimization, 41, 404-424, (2002).

10. Super-replication under Gamma constraints, (with N. Touzi) SIAM Journal on Control and Optimization, 39(1), 73-96, (2000).

·          Liquidity: Recently, together with Umut Çetin, we (Soner and Touzi) are applying the methods developed above to the liquidity model of Cetin-Jarrow-Protter.  Related papers are:

 

1.    Large liquidity expansion for super-hedging costs, (with U. Çetin and N. Touzi), 2008, in preparation.

2.    Option hedging for small investors under liquidity costs, (with U. Çetin and N. Touzi), 2006, Finance & Stochastics, submitted.

 

·         Second order Backward Stochastic Differential Equations:  Again based on our work on super-replication, together with Patrick Cheridito, Nizar Touzi and Nicolai Victoir we developed a generalization of backward stochastic differential equations.  This generalization enables to represent all fully nonlinear parabolic equations in terms of this new class of stochastic equations.  More recently, with Nizar Touzi and Takis Souganidis we are developing a weak-viscosity theory for the Markov case.  Also with Nizar Touzi and Jianfeng Zhang we generalized the probability space to enable us to have a satisfactory existence and duality theory for super-replication, stochastic target and backward stochastic differential equations.  We plan to work on the numerical aspects of this Project as well.  Related papers and preprints are:  

 

1.       Duality for 2BSDE’s and stochastic target problems, (with N. Touzi and J. Zhang), 2008, in preparation

2.       Stochastic representations for nonlinear parabolic PDEs, survey article,  (2007).

3.       .Second order backward stochastic differential equations and fully non-linear parabolic PDE's, (with P. Cheridito, N. Touzi, and N. Victoir), Comm. on Pure and Applied Math.,  60 (7): 1081-1110 (2007).

 

·         Merton problem with taxes: This line of research introduces a tractable model for capital gains taxes.  Two papers together with Imen Ben-Tahar and Nizar Touzi are:

 

1.     Merton problem with taxes: characterization, computation and approximation, (with I. Ben-Tahar and N.  Touzi),  2008, preprint (this is the revised version of the previous manuscript "Modeling continuous-time financial markets with capital gains taxes".)

2.    The dynamic programming equation for the problem of optimal investment under capital gains taxes, (with I. Ben-Tahar and N. Touzi),  SIAM Journal on Control and Optimization,  46 (5) : 1779-1801,  (2007).

 

·         Asymptotic theory for Ginzburg-Landau type equations:  Staring with scalar equations and then moving onto the systems, Bob Jerrard and I have developed techniques to study asymptotic problems for these problems.   Related papers are:

 

1.    Limiting behavior of the Ginzburg-Landau energy,  (with R.L. Jerrard) J. Functional Analysis, 192, 524-561, (2002).

2.    The Jacobian and the Ginzburg-Landau energy,  (with R.L. Jerrard) Calculus of Variations, 14 , 151-191, (2002). 

3.    Rectifiability of the distributional Jacobian for a class of functions, (with R.L. Jerrard) C.R. Acad. Sci. Paris, t. 329, Serie I, 983-688, (1999).

4.    Scaling limits and regularity for a class of Ginzburg-Landau systems, (with R.L. Jerrard) Annales L'Institute H. Poincare, 16/4, 423-466, (1999).

5.    Dynamics of Ginzburg-Landau vortices, (with R.L. Jerrard) Arc. Rat. Mech. An., 142, 185-206, (1998).

6.    Ginzburg-Landau equation and motion by mean curvature, I: convergence, Journal of Geometric Analysis, 7,437-475, (1997).

7.    Ginzburg-Landau equation and motion by mean curvature, II: development of the interface, Journal of Geometric Analysis, 7,476-491, (1997).

8.    Convergence of the phase field equations to the Mullins-Sekerka problem with a kinetic undercooling, Arc. Rat. Mech. An., 131, 139-197, (1995).

9.    Front propagation and phase field theory, (with G. Barles and P.E. Souganidis) SIAM J. Cont. Opt., 2/31, special issue dedicated to W. Fleming, 439-469, (1993).

10. Phase transitions and generalized motion by mean curvature, (with L.C. Evans and P.E. Souganidis) Comm. in Pure and Applied Math., 65, 1097-1123, (1992).


 

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last updated on November 25, 2008.