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Probability and Computational Finance Seminar
Tahir Choulli
University of Alberta
Title: Optional martingale representation for markets under random horizon with applications.

Abstract: The economic problem of how a random horizon impact investment, or the market in general, can be traced back to Irving Fisher (1931). However, there is little done in both mathematical and financial aspects. In this talk, I consider a general market model represented by its flow of information F and its underlying assets' price process S. Our random horizon is a random time that might represent the default time of a firm, the death time of an insured, or more generally an occurrence time of an event that might impact the market somehow. Thus, in general, one can not say whether T occurs or not using F only. Our resulting market model is the pair (S,F) over the stochastic interval [0,T]. By keeping in mind our interest in credit risk and life insurance, we model the new informational system by a larger flow of information G that incorporates both F and the information about T as it occurs. In this setting many questions arise. Among these, we ask the following: How viable is the new model? How one can ``measure" the impact of T on the optimal investment/portfolio/numeraire portfolio? What are the new risks that arise from the interplay between the uncertainties in F (or S) and T? Can we decompose G-risk into pure financial risk (F-risk), pure mortality/default/horizon risk, and correlation risk? Can we describe the set of all deflators for the new informational model? In the context of life insurance, the mortality and/or longevity risk and their securitization pose serious challenges, while the existing models for mortality are severely criticized. Herein, the question of how one can get the dynamics for mortality and/or longevity derivatives without mortality's specification, has a great importance! To answer the majority of these questions, we introduce new martingales and we elaborated an optional martingale representation result that describes the dynamics of any G-martingale stopped at T.

This talk is based on joint works with Catherine Daveloose, Michele Vanmaele (Ghent University), and Sina Yansori (UofA).

Date: Monday, February 19, 2018
Time: 4:30 pm
Location: Wean Hall 8220
Submitted by:  Dmitry Kramkov