In , Z. Chen proved that, if for each terminal condition $\xi$, the solution of the BSDE associated to the standard parameter $(\xi, g_1)$ is equal at time $t=0$ to the solution of the BSDE associated to $(\xi, g_2)$ then we must have $g_1\equiv g_2$. This result yields a natural question: what happens in the case of an inequality in place of an equality? In this paper, we try to investigate this question and we prove some properties of ``$g$-expectation'', notion introduced by S. Peng in .
"A Converse Comparison Theorem for BSDEs and Related Properties of g-Expectation." Electron. Commun. Probab. 5 101 - 117, 2000. https://doi.org/10.1214/ECP.v5-1025