Below is the standard result on existence and uniqueness of BSDE of the form

Notations follow the paper [Pen10].

- Let be a filtered probability space with -valued Brownian motion .
- Let be the collection of random variables satisfying .
- is the collection of the -adapted processes satisfying
- is the collection of the -adapted processes satisfying

Theorem 1([Pen10]) Suppose for all and . Then, for any , there exists unique .

Under the same setting, there is a similar result in

[YZ99] J. Yong and X. Y. Zhou. Stochastic controls: Hamiltonian systems and HJB equations, vol 43. Springer, 1999.

Theorem 2(Theorem 7.3.2 of [YZ99]) Suppose for all and . Then, for any , there exists unique .

It is obvious that is a subset of . Can we say they are equal?

NO. Consider following counter-example. Let be given by

Let , and . Set for all , and is generated by . Then, one can check