On $g$-expectations and filtration-consistent nonlinear expectations
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911787411046400 |
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| author | Zheng, Shiqiu |
| author_facet | Zheng, Shiqiu |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_07793 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On $g$-expectations and filtration-consistent nonlinear expectations Zheng, Shiqiu Probability 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. |
| title | On $g$-expectations and filtration-consistent nonlinear expectations |
| topic | Probability |
| url | https://arxiv.org/abs/2302.07793 |