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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2302.07793 |
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Table of Contents:
- In this paper, we obtain a comparison theorem and a invariant representation theorem for backward stochastic differential equations (BSDEs) without any assumption on the second variable $z$. Using the two results, we further develop the theory of $g$-expectations. Filtration-consistent nonlinear expectation (${\cal{F}}$-expectation) provides an ideal characterization for the dynamical risk measures, asset pricing and utilities. We propose two new conditions: an absolutely continuous condition and a (locally Lipschitz) domination condition. Under the two conditions respectively, we prove that any ${\cal{F}}$-expectation can be represented as a $g$-expectation. Our results contain a representation theorem for $n$-dimensional ${\cal{F}}$-expectations in the Lipschitz case, and two representation theorems for $1$-dimensional ${\cal{F}}$-expectations in the locally Lipschitz case, which contain quadratic ${\cal{F}}$-expectations.