Abstract
Given \(p \in (1,2]\), the wellposedness of backward stochastic differential equations with jumps (BSDEJs) in \(\mathbb {L}^p\) sense gives rise to a so-called g-expectation with \(\mathbb {L}^p\) domain under the jump filtration (the one generated by a Brownian motion and a Poisson random measure). In this paper, we extend such a g-expectation to a nonlinear expectation \(\mathcal{E}\) with \(\mathbb {L}^p\) domain that is consistent with the jump filtration. We study the basic (martingale) properties of the jump-filtration consistent nonlinear expectation \(\mathcal{E}\) and show that under certain domination condition, the nonlinear expectation \(\mathcal{E}\) can be represented by some g-expectation.
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Liu, J., Yao, S. Jump-Filtration Consistent Nonlinear Expectations with \({\mathbb {L}^{p}}\) Domains. Appl Math Optim 79, 87–129 (2019). https://doi.org/10.1007/s00245-017-9422-4
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DOI: https://doi.org/10.1007/s00245-017-9422-4
Keywords
- Backward stochastic differential equations with jumps
- \(\mathbb {L}^p\) solutions
- g-Expectations
- Nonlinear expectations consistent with jump filtration
- Optional sampling
- Doob–Meyer decomposition
- Representation theorem