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Existence for BSDE with superlinear-quadratic coefficient. (English) Zbl 0910.60046

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 \(z\).

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)

Citations:

Zbl 0880.60061
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