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