Representation theorems for generators of backward stochastic differential equations and their applications

We prove that the generator g of a backward stochastic differential equation (BSDE) can be represented by the solutions of the corresponding BSDEs at point ( t , y , z ) if and only if t is a conditional Lebesgue point of generator g with parameters ( y , z ) . By this conclusion, we prove that, if g is a Lebesgue generator and g is independent of y, then, Jensenʹs inequality for g-expectation holds if and only if g is super homogeneous; we also obtain a converse comparison theorem for deterministic generators of BSDEs.