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This paper is concerned with a kind of risk-sensitive optimal control problemfor fully coupled forward-backward stochastic systems. The control variableenters the diffusion term of the state equation and the control domain is notnecessarily convex. A new global maximum principle is obtained without assumingthat the value function is smooth. The maximum condition, the first- andsecond-order adjoint equations heavily depend on the risk-sensitive parameter.An optimal control problem with a fully coupled linear forward-backwardstochastic system and an exponential-quadratic cost functional is discussed.The optimal feedback control and optimal cost are obtained by using Girsanov'stheorem and completion-of-squares approach via risk-sensitive Riccatiequations. A local solvability result of coupled risk-sensitive Riccatiequations is given by Picard-Lindel$\ddot{o}$f's Theorem.

A Risk-Sensitive Global Maximum Principle for Controlled Fully Coupled FBSDEs with Applications