Standard result on existence and uniqueness of BSDE

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

\displaystyle   Y_{t} = Y_{T} + \int_{t} ^{T} g(s, Y_{s}, Z_{s}) ds - \int_{t}^{T} Z_{s} d W_{s}, \ Y_{T} = \xi. \ \ \ \ \ (1)

Notations follow the paper [Pen10].

  • Let {(\Omega, \mathcal F, \mathbb P, \{\mathcal F_{t}\}, W)} be a filtered probability space with {\mathbb R^{d}}-valued Brownian motion {W}.
  • Let {L^{p}_{\mathbb P} (\mathcal F_{t}, \mathbb R^{m})} be the collection of random variables {X\in \mathcal F_{t}/ \mathcal B(\mathbb R^{m})} satisfying {\mathbb E [ |X|^{p} ] < \infty}.
  • {M_{\mathbb P}^{p}(0,T, \mathbb R^{m})} is the collection of the {\mathcal F_{t}}-adapted processes {X: \Omega \times [0,T] \mapsto \mathbb R^{m}} satisfying

    \displaystyle \mathbb E [\int_{0}^{T} |X_{t}|^{p} dt] <\infty

  • {S_{\mathbb P}^{p}(0,T, \mathbb R^{m})} is the collection of the {\mathcal F_{t}}-adapted processes {X: \Omega \times [0,T] \mapsto \mathbb R^{m}} satisfying

    \displaystyle \mathbb E [\sup_{[0,T]} |X_{t}|^{p}] <\infty

Theorem 1 ([Pen10]) Suppose {g(\cdot, y, z) \in M_{\mathbb P}^{2}(0,T, \mathbb R^{m})} for all {T, y, z} and {g\in Lip_{y,z}}. Then, for any {\xi \in L_{\mathbb P}^{2}(\mathcal F_{T}, \mathbb R^{m})}, there exists unique {(Y,Z) \in M_{\mathbb P}^{2}(0,T, \mathbb R^{m}\times \mathbb R^{m\times d})}.

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 {g(\cdot, y, z) \in M_{\mathbb P}^{2}(0,T, \mathbb R^{m})} for all {T, y, z} and {g\in Lip_{y,z}}. Then, for any {\xi \in L_{\mathbb P}^{2}(\mathcal F_{T}, \mathbb R^{m})}, there exists unique {(Y,Z) \in S_{\mathbb P}^{2}(0,T, \mathbb R^{m}) \times M_{\mathbb P}^{2}(0,T, \mathbb R^{m\times d})}.

It is obvious that {S_{\mathbb P}^{2}(0,T, \mathbb R^{m})} is a subset of {M_{\mathbb P}^{2}(0,T, \mathbb R^{m})}. Can we say they are equal?

NO. Consider following counter-example. Let {\omega_{i} \in C[0,1]} be given by

\displaystyle \omega_{i} (t) = 0, \forall 0\le t < 1 - 2^{-i}, \quad \omega_{i}(t) = 2^{2i} (t - 1 + 2^{-i}), \forall 1-2^{-i} \le t \le 1.

Let {\Omega = \{\omega_{i} : i = 1, 2, \ldots\}}, and {\mathbb P\{ \omega_{i} \} = 2^{-i}}. Set {X(\omega, t) = \omega(t)} for all {\omega \in \Omega}, and {\mathcal F_{t}} is generated by {X}. Then, one can check

\displaystyle X\in M_{\mathbb P}^{2}(0,1, \mathbb R) \setminus S_{\mathbb P}^{2}(0,1, \mathbb R).

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