Backward stochastic differential equations with locally Lipschitz coefficient (Q2761878)

The author considers the multi-dimensional backward stochastic differential equation for \(t\in [0,1]\),NEWLINE\[NEWLINE Y_t=\xi+\int_t^1 f(s,Y_s,Z_s)ds-\int_t^1 Z_sdW_s.NEWLINE\]NEWLINE It is proved that this equation has a unique solution \((Y_t,Z_t)\) if 1) \(f\) is progressively measurable; 2) \(f\) is of sublinear growth: \(|f(t,y,z)|\leq M(1+|y|^\alpha+|z|^\alpha)\); 3) \(f\) has a local Lipschitz condition: \(|f(t,y,z)-f(t,y',z')|\leq L_N(|y-y'|+|z-z'|)\) for all \(|y|,|y'|,|z|,|z'|\leq N\) with \(L_N=\sqrt{(1-\alpha)\log(N)}\). Afterwards the result is generalized in two different ways. The main ingredients of the proof are truncation arguments and convergence results in a suitable Banach space.