Quadratic nonlinear expectations and risk measures

Song Yao, Purdue University

Abstract

In this thesis, we are aimed to relate dynamic risk measures more extensively to the backward stochastic differential equations (BSDEs) in the quadratic case. We first extend the notion of g-evaluation, in particular g-expectation, to the quadratic case and study some of its important properties. Next we extend the notion of “filtration-consistent nonlinear expectation” ([special characters omitted]-expectation) to the case when it is allowed to be dominated by quadratic g-expectation. One of our main contributions is to replace the domination condition used in the lipschitz case (e.g., [9] and [31]) with a new one that is appropriate for the quadratic case. Then we show that [special characters omitted]-expectations with the new domination have many fundamental martingale properties including the Doob-Meyer type decomposition theorem and the optional sampling theorem. The thesis eventually highlights in a representation result: Any dynamic risk measure with the new domination must be a quadratic g-expectation, or the solution to a quadratic BSDE.

Degree

Ph.D.

Advisors

Ma, Purdue University.

Subject Area

Mathematics

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