A counter-example on the uniqueness of Mean field FBSDE

We copy an interesting simple FBSDE with its many explicit solutions from [CD13].

Consider a system of equations with unknown triple {(X, Y, Z)} given by

\displaystyle  dX_{t} = \mathbb E[Y_{t}] dt + dW_{t}, \quad X_{0} = 0;

and

\displaystyle  d Y_{t} = - \mathbb E [X_{t}] dt + Z_{t} dW_{t}, \quad Y_{\pi/4} = \mathbb E[X_{\pi/4}].

One can easily check that, for any real number {a}, it has the solution of the form

\displaystyle X_{t} = a \sin t + W_{t}, \quad Y_{t} = a \cos t, \quad Z_{t} = 0; \quad t\in [0, \pi/4].

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