# Sufficient condition for near-optimal control of general controlled linear forward–backward stochastic differential equations

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## Abstract

This article studies sufficient conditions for near-optimal stochastic control for systems governed by general linear controlled forward–backward stochastic differential equations (FBSDEs in short). The control is allowed to enter into both drift and diffusion coefficients. We prove that under certain additional conditions on the Hamiltonian, the near-maximum condition on the Hamiltonian function in the integral form is sufficient for near-optimality. As an applications, an example is given to illustrate our theoretical results.

### Keywords

Sufficient conditions for near-optimal control Forward–backward stochastic differential equations Second-order adjoint equations Ekeland’s variational principle Clarke’s generalized gradient### Mathematics Subject Classification

93E20 60H10### References

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