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This book consists of four chapters. In first chapter there is a short review of theory of Backward Stochastic Differential Equations (BSDEs) and Bounded Mean Oscillation (BMO) martingales. In second chapter an interesting connections between theory of BSDEs and BMO martingales is studied. Using the BSDE tool a new proofs of some classical results on BMO martingales are provided. In Third chapter we have studied Backward Stochastic Differential Equations with a convex generator of quadratic growth. Existence and uniqueness of a solution is proved for such equations driven by continuous martingale with unbounded characteristic. Results on the existence and uniqueness for BSDEs with quadratic growth we have used in fourth chapter, to solve the linear-quadratic regulator (LQR) problem in general martingale setting. We derived the corresponding BSDE for LQR problem and expressed the optimal strategy of LQR problem in terms of the unique solution of corresponding BSDE.