The uniqueness of solutions of perturbed backward stochastic differential equations (Q1765781)

The author studies existence and uniqueness for one-dimensional backward stochastic differential equations driven by a one-dimensional Brownian motion. The coefficients, only dependent on \(y\), are supposed to be continuous functions \(f:R\rightarrow R\) for which there exists some nondecreasing continuous function \(h:R_+\rightarrow R_+\) with \(h(0)=0,h(u)>0\) for \(u>0\) and \(\int_{0+}^1 uh^{-2}(u)du=+\infty\) such that \(| f(y_1)-f(y_2)|\leq h(| y_1-y_2|),\, y_1,y_2\in R\), while this assumption is slightly modified for the perturbating coefficients. Let us recall that, in the case of one-dimensional BSDE driven by a multi-dimensional Brownian motion, it is known that the BSDE has a solution if the driven coefficient is an adapted process depending continuously on \((y,z)\) and satisfying some growth assumption [see \textit{J. P. Lepeltier} and \textit{J. San Martin}, Stat. Probab. Lett. 32, No. 4, 425--430 (1997; Zbl 0904.60042) and Stochastics Stochastics Rep. 63, No. 3--4, 227--240 (1998; Zbl 0910.60046)]. In the multi-dimensional case, one of the most recent existence and uniqueness results was obtained by \textit{K. Bahlali, E. H. Essaky, M. Hassani} and \textit{E. Pardoux} [C. R., Math., Acad. Sci. Paris 335, No. 9, 757--762 (2002; Zbl 1017.60068)]. With their local monotonicity assumption the coefficient can be neither locally Lipschitz in \(y\) nor in \(z\).