Backward stochastic differential equations with a convex generator (Q2882539)

Let \((\Omega,{\mathcal F},({\mathcal F}_t)\,(0\leq t\leq T), \operatorname{P})\) be a filtered probability space such that every \(({\mathcal F}_t)\)-local martingale is continuous. The author studies a backward stochastic differential equation (BSDE) of the form NEWLINE\[NEWLINEY_t= Y_0- \int^t_0 f(s,Z_s)\,d\langle M\rangle_s+ \int^t_0 Z_s dM_s+ L_t,\quad Y_T= \eta,\tag{\(*\)}NEWLINE\]NEWLINE where the generator \(f: [0,T]\times \Omega\times\mathbb{R}\to \mathbb{R}\) is measurable and, for any \(z\), \(f(\cdot,\cdot,z)\) is predictable, \(\eta\) is an \({\mathcal F}_T\)-measurable random variable, and \((M_t)\) \((0\leq t\leq T)\) is a given square integrable \(({\mathcal F}_t)\)-martingale. A solution of \((*)\) is a triple \((Y,Z,L)\), where \((Y_t)\) \((0\leq t\leq T)\) is a semimartingale, \((Z_t)\) \((0\leq t\leq T)\) is predictable such that NEWLINE\[NEWLINE\operatorname{E}\Biggl[\int^T_0 Z^2_s d\langle M\rangle_s\Biggr]< \infty,NEWLINE\]NEWLINE and \((L_t)\) \((0\leq t\leq T)\) is a square integrable martingale orthogonal to \(M\) such that \((Y,Z,L)\) satisfies \((*)\). Let \({\mathcal E}_{t,s}(\int f_l'(u)\,dM)\) denote the unique solution of the linear SDE NEWLINE\[NEWLINEdX_s= X_sf_l'(s, u_s)\,dM_s,\quad X_t= 1\;(t\leq s\leq T)NEWLINE\]NEWLINE (\(f_l'\) denoting the left derivative of \(f\) w.r.t. \(z\)). Let BMO denote the class of all continuous martingales \(M\) such that NEWLINE\[NEWLINE\sup_\tau \| \operatorname{E}[\langle M\rangle_T- \langle M\rangle_\tau\mid F_\tau]\|_\infty< \inftyNEWLINE\]NEWLINE (the supremum taken over all stopping times \(0\leq \tau\leq T\)). The main result is the following.NEWLINENEWLINE Let \((M_t)\in\text{BMO}\) and assume that \(f(t,\omega,\cdot)\) is continuous, convex and satisfies a certain quadratic growth condition. Then (under some additional technical assumption) there exists a solution \((V_t)\) \((0\leq t\leq T)\) of \((*)\) of the form NEWLINE\[NEWLINEV_t= \text{ess\,sup}_{u\in U}\operatorname{E}\Biggl[\operatorname{E}_{t,T}(\int f_l'(u)\,dM)\Biggl(\eta+ \int^T_t [f(s, u_s)- f_l'(s, u_s)\,u_s]\,d\langle M\rangle_s\Biggr)\Biggl|{\mathcal F}_t\Biggr],NEWLINE\]NEWLINE where \(U\) is the class of predictable bounded controls.