Generalized fractional smoothness and \(L_p\)-variation of BSDEs with non-Lipschitz terminal condition (Q424522)

The paper presents an interesting result on the solution of the backward stochastic differential equation NEWLINE\[NEWLINEY_t = \xi + \int_t^T f(s,X_s,Y_s,Z_s)\,ds - \int_t^T Z_s dW_s, \quad t\in [0,T]\;\; a.s.NEWLINE\]NEWLINE with a Lipschitz generator \(f\) and a terminal condition \(\xi=g(X_{\tau_1},\dotsc, X_{\tau_L})\) with \(0=\tau_0<\dotsb<\tau_L=T\), where \(g\) is not necessarily a Lipschitz function.NEWLINENEWLINEAfter a good introduction on the concept of backward fractional smoothness of a backward stochastic differential equation and some characterizations of this smoothness, the authors use some results given by the first two authors [Stochastics Stochastics Rep. 76, No. 4, 339--362 (2004; Zbl 1060.60056)] and by the third author and \textit{A. Makhlouf} [Stochastic Processes Appl. 120, No. 7, 1105--1132 (2010; Zbl 1195.60079)] to prove two sufficient conditions for smoothness (Corollary 2 and Theorem 4). The point of these two conditions is that they only involve the terminal condition \(\xi\) and do not use the solution \(Y\) nor the generator \(f\) of this BSDE. The proofs of the main results are presented in the third section.NEWLINENEWLINETheir results extend some others results on the frame of backward stochastic differential equations as for example there given by \textit{B. Bouchard} and \textit{N. Touzi} [Stochastic Processes Appl. 111, No. 2, 175--206 (2004; Zbl 1071.60059)], \textit{J. Zhang} [Ann. Appl. Probab. 14, No. 1, 459--488 (2004; Zbl 1056.60067)], the third author, \textit{J.-Ph. Lemor} and \textit{X. Warin} [Ann. Appl. Probab. 15, No. 3, 2172--2202 (2005; Zbl 1083.60047); Bernoulli 12, No. 5, 889--916 (2006; Zbl 1136.60351)].