Weak solutions of backward stochastic differential equations with continuous generator

Nadira Bouchemella (1) , Paul Raynaud de Fitte (1)

1 LMRS - Laboratoire de Mathématiques Raphaël Salem

We prove the existence of a weak solution to a backward stochastic differential equation (BSDE) $$ Y_t=\xi+\int_t^T f(s,X_s,Y_s,Z_s)\,ds-\int_t^T Z_s\,d\wien_s$$ 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 $$ Y_t=\xi+\int_t^T f(s,X_s,Y_s,Z_s)\,ds-\int_t^T Z_s\,d\wien_s-(L_T-L_t)$$ where $L$ is a continuous martingale which is orthogonal to any $\wien$. 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.

Mots clés: Young measure; Skorokhod space; Jakubowski's topology S; condition UT; Meyer-Zheng; pathwise uniqueness; Yamada-Watanabe-Engelbert; Backward stochastic differential equation; weak solution; martingale solution; joint solution measure

Domaines: Probabilités [math.PR]

Nadira Bouchemella, Paul Raynaud de Fitte. Weak solutions of backward stochastic differential equations with continuous generator. Stochastic Processes and their Applications, 2014, 124 (1), pp.927-960. ⟨10.1016/j.spa.2013.09.011⟩. ⟨hal-00583676v4⟩