Lepeltier, J.-P.; San Martín, J. Existence for BSDE with superlinear-quadratic coefficient. (English) Zbl 0910.60046 Stochastics Stochastics Rep. 63, No. 3-4, 227-240 (1998). Consider a one-dimensional backward stochastic differential equation of the form \[ Y_t= \xi+\int^T_t f(s,w,Y_s, \Lambda_s) ds- \int^T_t \Lambda_s dW_s, \] where \(W_t\) is a \(d\)-dimensional Brownian motion and the terminal variable \(\xi\) is bounded. A solution is a pair of predictable and square integrable processes \((Y_t, \Lambda_t)\). The coefficient \(f(t,w,y,z)\) is supposed to be continuous in \((y,z)\) and predictable. It is proved that if \(f\) satisfies a growth condition of the form \(| f(t,w,y,z) |\leq l(y) +C| z|^2\), where \(l\) is any strictly positive function such that \(\int^\infty_0 {dx \over l(x)} =\int^0_{-\infty} {dx\over l(x)} =\infty\), then the above equation has a maximal bounded solution. This result extends a previous work of the authors [Stat. Probab. Lett. 34, 347-354 (1997)] where \(f\) is supposed to have linear growth in \(y\) and \(z\), and also generalizes a result by M. Kobylanski [C. R. Acad. Sci., Paris, Sér. I 324, No. 1, 81-86 (1997; Zbl 0880.60061)] who considered the case when \(f\) has linear growth in \(y\) and quadratic growth in \(z\). Reviewer: D.Nualart (Barcelona) Cited in 3 ReviewsCited in 60 Documents MSC: 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) Keywords:backward stochastic differential equations; growth condition; maximal bounded solution Citations:Zbl 0880.60061 PDFBibTeX XMLCite \textit{J. P. Lepeltier} and \textit{J. San Martín}, Stochastics Stochastics Rep. 63, No. 3--4, 227--240 (1998; Zbl 0910.60046) Full Text: DOI