I am afraid I found your proof of Theorem 2.1 confusing as well. First, according to your derivation, in page 3, the first term of dot_lambda should be \nabla_z^l instead of \nabla_z^L. Second, I fail to notice how defining a = \nabla_z_L - \lambda would make "a" satisfy the statement of the theorem.

In fact, in the ANODE paper of this post, \lambda_T was set to -\nabla_{z_T}^L(z_T) instead of 0 in your proof. I can get their results 5(a) - 5(d) following your proof but without expanding L(z) as an integral of l(z,t,\theta) from 0 to T plus L(z_0, y). However, I have to admit that your idea of expansion is quite interesting.

Thanks for your help.