Representation theorem and viability property for multidimensional BSDEs and their applications

The representation theorem and the viability property for backward stochastic differential equations (BSDEs) require further exploration, given their widespread use in both theory and practical applications. In this study, we present a positive answer to the long-standing open question of whether the representation theorem still holds in the $ L^2 $ -sense under the standard assumptions of square integrability and Lipschitzian continuity on the generators of BSDEs. In the process, the multidimensional case is considered. Subsequently, based on the representation theorem, we obtain a necessary and sufficient condition for the viability property of the BSDEs under standard conditions on the generators. This removes the requirement for the generator to possess the properties of stronger integrability and continuity with respect to time variables. As an application of these results, we conduct various types of comparisons and converse comparisons for the solutions of multidimensional BSDEs, and several properties of the multidimensional $ g $ -expectation are obtained.