We obtain a representation theorem for the generators of BSDEs driven by $G$-Brownian motions and then we use the representation theorem to get a converse comparison theorem for $G$-BSDEs and some equivalent results for nonlinear expectations generated by $G$-BSDEs.

We get the exponential G-martingale theorem with the Kazamaki condition and tell a distinct difference between the Kazamaki’s and Novikov’s criteria with an example.

We investigate the tailed asymptotic behavior of the randomly weighted sums with increments with convolution-equivalent distributions. Our obtained result can be directly applied to a discrete-time insurance risk model with insurance and financial risks and derive the asymptotics for the finite-time probability of the above risk model.

For a mixed stochastic Volterra equation driven by Wiener process and fractional Brownian motion with Hurst parameter $H>1/2$, we prove an existence and uniqueness result for this equation under suitable assumptions.