Weak solutions of backward stochastic differential equations with continuous generator

We prove the existence of a weak solution to a backward stochastic differential equation (BSDE) Y t = ξ + ∫ t T f ( s , X s , Y s , Z s ) d s − ∫ t T Z s d W s in a finite-dimensional space, where f ( t , x , y , z ) is affine with respect to z , and satisfies a sublinear growth condition and a continuity condition. This solution takes the form of a triplet ( Y , Z , L ) of processes defined on an extended probability space and satisfying Y t = ξ + ∫ t T f ( s , X s , Y s , Z s ) d s − ∫ t T Z s d W s − ( L T − L t ) where L is a martingale with possible jumps which is orthogonal to W . The solution is constructed on an extended probability space, using Young measures on the space of trajectories. One component of this space is the Skorokhod space D endowed with the topology S of Jakubowski.