Convexity and sublinearity of \(g\)-expectations. (English) Zbl 1498.60221
Summary: Under the basic assumptions of \(g\)-expectations defined in [Z. Chen and B. Wang, J. Aust. Math. Soc., Ser. A 69, No. 2, 187–211 (2000; Zbl 0982.60052)], we establish the one-to-one correspondence between generators of backward stochastic differential equations (BSDEs for short) and the convexity (resp. conditional convexity, \( \mathcal{F}_t\)-convexity) of \(g\)-expectations, respectively. We also obtain the relationship between generators of BSDEs and the sublinearity (resp. conditional sublinearity, \( \mathcal{F}_t\)-sublinearity) of \(g\)-expectations, respectively. Moreover, we provide the reasonable assumptions on generators of BSDEs for the further study on dynamic risk measures by applying the theory of \(g\)-expectations.
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 91G05 | Actuarial mathematics |
| 60H30 | Applications of stochastic analysis (to PDEs, etc.) |
Keywords:
convexity; sublinearity; backward stochastic differential equation; \(g\)-expectation; dynamic risk measureCitations:
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