On \(g\)-evaluations with \(\mathbb{L}^p\) domains under jump filtration. (English) Zbl 1382.60085
Summary: Given \(p \in (1, 2)\), the unique \(\mathbb{L}^p\) solutions of backward stochastic differential equations with jumps (BSDEJs) allow us to extend the notion of \(g\)-evaluations, in particular \(g\)-expectations, to the jump case with \(\mathbb{L}^p\) domains. We explore many important properties of the extended \(g\)-evaluations, including optional sampling, upcrossing inequality, Doob-Meyer decomposition, generator representation, and Jensen’s inequality. Most of these results are important for the further development of jump-filtration consistent nonlinear expectations with \(\mathbb{L}^p\) domains in [S. Yao and J. Liu, “Jump-filtration consistent nonlinear expectations with \(\mathbb{L}^p\) domains”, Appl. Math. Optim. (to appear)].
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 60J75 | Jump processes (MSC2010) |
| 60F25 | \(L^p\)-limit theorems |
Keywords:
backward stochastic differential equation with jumps; \(\mathbb{L}^p\) solutions; \(g\)-evaluation; \(g\)-expectation; optional sampling; upcrossing inequality; Doob-Meyer decomposition; generator representation; reverse comparision theorem of BSDEJs; Jensen inequalityReferences:
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