Markovian quadratic and superquadratic BSDEs with an unbounded terminal condition (Q444352)

Existence and uniqueness is proved for Markovian quadratic and superquadratic backward stochastic differential equations (BSDE) of the form NEWLINE\[NEWLINEY_t= g(X_T)+ \int^T_t f(s, X_s, Y_s, Z_s)\,ds- \int^T_0 JZ_s dW_s,NEWLINE\]NEWLINE where \(X_t\) is the solution of the stochastic differential equation NEWLINE\[NEWLINEX_t= x+ \int^t_0 b(s, X_s)\,ds+ \int^t_0 \sigma(s)\,dW_s,NEWLINE\]NEWLINE \(W_t\) is a \(d\)-dimensional Brownian motion, \(f\) has quadratic or superquadratic growth with respect to \(z\), and \(\sigma\) is deterministic. Then, the case where \(\sigma\) is random is considered, and, under more restrictive conditions, existence, uniqueness, and boundedness of \(Z\) is established. Applications to semilinear partial differential equations are explored. The paper concludes by studying the approximation of the ESDE and establishing error bounds on numerical approximations obtained using the Euler method.