# 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

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