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G-Expectations and Nonlinear Martingales

Organizers Marcel Nutz, H. Mete Soner
Place HG F 26.5
Time We 10-12
Beginning Feb 23
Contact Marcel Nutz
Prerequisites Solid knowledge of continuous stochastic processes.
Description This seminar is aimed at Master's students (or PhD students) interested in probability theory, stochastic processes, stochastic optimal control and/or mathematical finance. We study selected topics related to so-called G-expectations, which advance ideas from backward stochastic differential equations (BSDEs) and the associated "g-expectations". The main difference to the latter is that G-expectations involve families of singular (rather than equivalent) probability measures. From a PDE point of view, this causes the problems to be "fully nonlinear".

The G-expectations (and more general nonlinear expectations) give rise to the notion of a nonlinear martingale. This is related to dynamic asset pricing and risk measurement in financial applications as well as to probabilistic representations of (fully nonlinear) parabolic PDEs in analysis.

The topics of the seminar have been very active fields of research in recent years and might be of interest for a Master's/PhD thesis.

 Organization In the first meeting, I (Marcel) will give an overview of the field and the suggested topics.

In a later meeting, after you looked at your assigned paper, we will discuss briefly the contents of your talk, which you will then prepare in detail.

One week before your talk, you hand in a "handout" with the most important definitions and theorems in your talk.

Registration We welcome registrations before the first meeting (email to Marcel) and may even hand out some topics on that basis. We intend to distribute most of the topics after the first meeting.
Additional information Each student is expected to give a talk about (a part of) one of the above papers. Topics will be assigned in more detail in the first meeting. Talks are in English.

The seminar yields 6 credit points. To obtain these, students should regularly attend the seminar and give a successful talk.

Literature (preliminary) NOTE: You may need to use an IP address from ETH (computer at ETH or via VPN) to access journal papers. Most of the documents below correspond to more than one assignment - don't panic.

[P] S. Peng. Nonlinear expectations and stochastic calculus under uncertainty. Preprint arXiv:1002.4546v1, 2010. Link (this is a book)

[Pe] S. Peng. Tightness, weak compactness of nonlinear expectations and application to CLT. Preprint arXiv:1006.2541v1, 2010. Link

[NS] M. Nutz and H. M. Soner. Superhedging and dynamic risk measures under volatility uncertainty. Preprint arXiv:1011.2958v1, 2010. Link

[STZ] H. M. Soner, N. Touzi, and J. Zhang. Martingale representation theorem for the G-expectation. Stochastic Process. Appl., 121(2):265-287, 2011. Link

[S] Y. Song. Some properties on G-evaluation and its applications to G-martingale decomposition. Preprint arXiv:1001.2802v2, 2010. Link

[DHP] L. Denis, M. Hu, and S. Peng. Function spaces and capacity related to a sublinear expectation: application to G-Brownian motion paths. To appear in Potential Anal., 2010. Link

[N] M. Nutz. Random G-expectations. Preprint arXiv:1009.2168v1, 2010. Link

 

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© 2014 Mathematics Department | Imprint | Disclaimer | 23 February 2011
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