Connection between the Adjoint Variables and Value Function for Controlled Fully Coupled FBSDEs: The Local Case

This paper is concerned with the connection between the local maximum principle and the dynamic programming principle for controlled fully coupled forward-backward stochastic differential equations (FBSDEs) when the control domain is convex and the diffusion term does not contain the variable z. The relation among the adjoint variables and the value function is obtained within the viscosity solution framework. The first-order adjoint equation of [M. S. Hu, S. L. Ji and X. L. Xue, A global stochastic maximum principle for fully coupled forward-backward stochastic systems, arXiv: 1803.02109] can be derived from the adjoint FBSDE of [Z. Wu, Maximum principle for optimal control problem of fully coupled forward-backward stochastic systems, Systems Science and Mathematical Sciences, vol. 11, 1998, pp. 249-259], by our main theorem. Additionally, we give an example to show the applications of the theoretical result.