Path regularity for solutions of backward stochastic differential equations (Q1601801)

The authors study a class of backward stochastic differential equations (BSDE) whose terminal values are functionals of a (forward) diffusion process, more precisely, they investigate the BSDE \(dY_t = -f(t, X_t, Y_t, Z_t) dt+\langle Z_t,dW_t\rangle\), \(Y_T= \xi \), where \(X_T\) is a diffusion process given by \(dX_t = b(t,X_t) dt + \sigma(t, X_t) dW_t\), \(X_0=x\), and \(\xi\) a functional of the process \(X_t\), \(t \in [0,T]\), with adapted solution \((X_t,Y_t,Z_t)\). They show that the conjecture on the path regularity of \(Z_t\), the martingale part of the solution, holds true: Under the conditions that the terminal value \(\xi\) is Lipschitz either in the \({\mathbf L}^{\infty}\)-norm or the \({\mathbf L}^1\)-norm and the coefficients satisfy minimal Lipschitz conditions, the process \(Z_t\) has at least a càdlàg version. The proof is based on a previous result by the authors about path regularity of \(Z_t\) in case that the terminal condition depends on only finitely many points of \(X_t\), that is \(\xi = g(X_{t_1},\dots,X_{t_n})\), and an approximation scheme in the Meyer-Zheng topology of pseudo-paths.