Note on stochastic control problems related with general fully coupled forward-backward stochastic differential equations

In this paper we study stochastic optimal control problems of general fully coupled forward-backward stochastic differential equations (FBSDEs). In Li and Wei [8] the authors studied two cases of diffusion coefficients $σ$ of FSDEs, in one case when $σ$\ depends on the control and does not depend on the second component of the solution $(Y, Z)$ of the BSDE, and in the other case $σ$ depends on $Z$ and doesn't depend on the control. Here we study the general case when $σ$ depends on both $Z$ and the control at the same time. The recursive cost functionals are defined by controlled general fully coupled FBSDEs, then the value functions are given by taking the essential supremum of the cost functionals over all admissible controls. We give the formulation of related generalized Hamilton-Jacobi-Bellman (HJB) equations, and prove the value function is its viscosity solution.

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