Abstract
We consider a diffusion control problem where the controller totally determines the state's diffusion coefficient but has no influence on the state's drift rate. By using the Pontryagin maximum principle we characterize an optimal control in terms of the adjoint forward-backward stochastic differential equation (FBSDE), turning out to be fully coupled. We use the method of decoupling fields for proving that the adjoint FBSDE possesses a solution.
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