Before revision:  To solve this problem, we provide a connection between the dynamic convex risk functional introduced and the solutionof a quadratic RBSDE (Reflected Backward Stochastic Differential Equations): this is achieved by studying the properties of specific non linear expectations  .
After revision:  To solve this problem, we provide a new characterization of the solutions of specific reflected backward stochastic differential equations (or RBSDEs) whose driver g is convex and has quadratic growth in its second variable: this is done by introducing the extended notion of g-Snell enveloppe. Then, in a second step, we relate this representation to a specific class of dynamic monetary concave functionals already introduced in a discrete time setting. This connection implies that the solution, characterized by means of non linear expectations .