portrait mete soner
Last updated: 05.10.2016
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RESEARCH INTERESTS

Soner has worked on stochastic optimal control, nonlinear partial differential equations and mathematical finance. His work on viscosity solutions, asymptotic analysis of Ginzburg-Landau equations and the Cahn-Allen equation and weak solution of parabolic geometric equations have received attention. Related papers can be found in the publications section.

Below is an old (from 2013) summary of the research projects currently pursued in the group. The research highlights of Soner's group during the period 2011 to 2016 is briefly outlined in this short research summary.

In addition to resources of the Chair, these research activities are also supported by the European Research Council through an advanced research grant FiRM, Swiss Finance Institute, Swiss National Science Foundation and by the ETH Foundation.

Robust Hedging and Martingale Optimal Transport

This research explores model independent bounds for prices of path-dependent options. The financial market consist of a dynamically traded stock and statically held options. General duality results have been proved in joint work with Yan Dolinsky from Hebrew University

Related recent papers in this direction are:

  1. Martingale optimal transport and robust hedging in continuous time, (with Y. Dolinsky), submitted to Probability Theory and Related Fields, (2013).
  2. Robust hedging with proportional transaction costs, (with Y. Dolinsky), (2013), preprint.

Small Transaction Costs

We study portfolio management problems in a market with small transaction costs. We use techniques from homogenization theory to obtain asymptotic formulae for the impact of the transaction cost both on the performance and the trading strategies.

Related recent papers in this direction are:

  1. Homogenization and asymptotics for small transaction costs, (with N. Touzi), SIAM Journal on Control and Optimization, (2012), forthcoming.
  2. Homogenization and asymptotics for small transaction costs - the multi-dimensional case, (with D. Possamai and N. Touzi), (2013), preprint.
  3. Asymptotics with fixed transaction costs, (with A. Altravici and J. Muhle-Karbe), working paper, (2013).

Second order BSDEs and Stochastic target Problems

This research is initially motivated by the connection between semilinear partial differential equations and backward stochastic equations as proved by Pardoux and Peng. Our goal was to extend this stochastic representation to all fully nonlinear parabolic equations. Indeed, in joint work with Cheredito, Touzi and Victoir, we extended this connection to a new class of backward stochastic equations. It then became clear that a new formulation using a set of singular measures instead of one fixed probability measure is better suited for this project. Then, in joint work with Touzi and Zhang, we developed this approach and proved uniqueness and an existence result as well as a duality result for stochastic target problems.

Related recent papers in this direction are:

  1. 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).
  2. Dual Formulation of Second Order Target Problems, (with N. Touzi and J. Zhang), Annals of Applied Probability, (2013).
  3. Wellposedness of Second Order Backward SDEs, (with N. Touzi and J. Zhang), Probability Theory and Related Fields, 153, 149 -- 190, (2012).
  4. Quasi-sure stochastic analysis through aggregation, (with N. Touzi and J. Zhang), Electronic Journal of Probability, (Article Number: 67), 16, 1844--1879, (2011).

Uncertain Volatility: Superhedging, Nonlinear expectations and Dynamic Risk Measures

We study financial markets in which volatility is not precisely known. The analysis of this risk factor introduces new and interesting mathematical challenges. This project is also intimately connected to the previous one. If in the 2BSDE, one focuses on a special nonlinearity then resulting problems are precisely the hedging and pricing equations for markets with uncertain volatility. The G-expectations of Peng and the quasi-sure analysis of Denis & Martini are also very closely related to this project.

Related recent papers in this direction are:

  1. Martingale Representation Theorem for the G-expectation, (with N. Touzi and J. Zhang), Stochastic Processes and their Applications,121, 265-287, (2011).
  2. Superhedging and Dynamic Risk Measures under Volatility Uncertainty, (with M. Nutz), SIAM Journal of Control and Optimization, xxx (2012).
  3. Random G-Expectations, (M. Nutz), preprint, 2010.
  4. Weak Approximation of G-Expectations, (with Y. Dolinsky and M. Nutz), 122 (2), Stochastic Processes and their Applications, 664--675, (2012).
Models for illiquidity

In this project we try to develop mathematically tractable models for liquidity. According to Kyle (1985) there are three important components of of illiquidity: depth, tightness and resilience. Our goal is to develop tractable mathematical phenomenological models in continuous time that incorporate all these aspects. We also develop the mathematical tools to analyze their properties.

Related recent papers in this direction are:

  1. Option hedging for small investors under liquidity costs, (with U. Çetin and N. Touzi), Finance & Stochastics, 14(3), 317-341, (2010).
  2. Liquidity in a Binomial market, (with S. Gokay), Mathematical Finance, 22/2, 250--276, (2012).
  3. Liquidity Models in Continuous and Discrete Time, (with S. Gokay and A.F. Roch), in Advanced Mathematical Methods for Finance , editors G. Di Nunno and B. Oksendal, Springer-Verlag, 333--366, (2011).
  4. Large liquidity expansion for super-hedging costs, (with D. Possamai and N. Touzi), Asymptotic Analysis, (2012).
  5. Utility maximization in an illiquid market, (with M. Vukelja), Stochastics - special issue in memory of M. Taksar, (2013), forthcoming.