\[ \left( v ^ { \alpha _ { n } } ( x ) - v ^ { \alpha _ { n } } ( x _ { 0 } ) \right) \to v ( x ) \]

for all \( x \in \mathcal { X } \), for some \( \lambda \in \mathbb { R } \) and some function \( v : \mathcal { X } \rightarrow \mathbb { R } \). Note that \|λ\|and \|v(·)\| are both bounded by \( C ^ { \prime } \). Finally, we notice that, for any \( x \in \mathcal { X } \),

\[ \alpha _ { n } v ^ { \alpha _ { n } } ( x ) = \alpha _ { n } v ^ { \alpha _ { n } } ( x _ { 0 } ) + \alpha _ { n } \bigl ( v ^ { \alpha _ { n } } ( x ) - v ^ { \alpha _ { n } } ( x _ { 0 } ) \bigr ) \to \lambda , \]

so the convergence of this sequence to \( \lambda \) holds for all x.

We are now in a position to prove existence of solutions to our Ergodic BSDE.

Theorem 4.4. Let v and \( \lambda \) be as constructed in Lemma 4.3. The triple \( ( Y , Z , \lambda ) \), where

\[ Y _ { t } : = v ( X _ { t } ) , \qquad e _ { k } ^ { * } Z _ { t } : = v ( e _ { k } ) . \]

is the unique bounded, stationary (i.e. does not depend on t), Markovian solution, with \( v ( x _ { 0 } ) = 0 \), to the Ergodic BSDE

\[ Y _ { T } = Y _ { t } - \sum _ { t \le u < T } \left( f ( X _ { u } , Z _ { u } ) - \lambda \right) + \sum _ { t \le u < T } Z _ { u } ^ { * } M _ { u + 1 } . \eqno ( 4 . 4 ) \]

Any other bounded solution \( ( Y ^ { \prime } , Z ^ { \prime } , \lambda ^ { \prime } ) \) satisfies \( \lambda \ = \ \lambda ^ { \prime } \), and any other bounded, stationary, Markovian solution \( ( \dot { Y } ^ { \prime } , Z ^ { \prime } , \lambda ^ { \prime } ) \) satisfies \( Y _ { t } \, = \, Y _ { t } ^ { \prime } + c \) for some \( c \in \mathbb { R } \), and \( Z \sim _ { M } Z ^ { \prime } \) 0.

Proof. Let \( \{ \alpha _ { n } \} _ { n > 1 } \) be the sequence constructed in Lemma 4.3. We have that \( Y _ { t } ^ { \alpha _ { n } } = v ^ { \alpha _ { n } } ( X _ { t } ) \) and \( e _ { k } ^ { * } Z _ { t } ^ { \alpha _ { n } } = v ^ { \alpha _ { n } } ( e _ { k } ) \) solve the discounted BSDE

\[ Y _ { T } ^ { \alpha _ { n } } = Y _ { t } ^ { \alpha _ { n } } - \sum _ { t \leq u < T } \left( f ( X _ { u } , Z _ { u } ^ { \alpha _ { n } } ) - \alpha _ { n } Y _ { u } ^ { \alpha _ { n } } \right) + \sum _ { t \leq u < T } \left( Z _ { u } ^ { \alpha _ { n } } \right) ^ { * } M _ { u + 1 } . \]

However, since \( \| \mathbf { 1 } \| _ { M _ { t + 1 } } = 0 \), and f does not distinguish between values of \( Z _ { u } \) up to equivalence \( { \sim } { M } _ { u + 1 } \) , we can equally write \( Z _ { u } ^ { \alpha _ { n } } - v ^ { \alpha _ { n } } ( x _ { 0 } ) \mathbf { 1 } \) in the place of \( Z _ { u } ^ { \alpha _ { n } } \) in the above. Note that \( e _ { k } ^ { * } Z _ { u } ^ { \alpha _ { n } } - v ^ { \alpha _ { n } } ( x _ { 0 } ) \to e _ { k } ^ { * } Z _ { u } \) as \( n \to \infty \) for each \( e _ { k } \in \mathcal { X } \), and that, by the bound established in Lemma 4.3, \|e∗kZuαn −vαn(x0)\|is uniformly bounded. Since f is Lipschitz in z, we deduce that

\[ f ( X _ { u } , Z _ { u } ^ { \alpha _ { n } } - v ^ { \alpha _ { n } } ( x _ { 0 } ) \mathbf { 1 } ) \to f ( X _ { u } , Z _ { u } ) \quad { \mathrm { a s } } \quad n \to \infty \quad { \mathrm { a . s . } } \]

It follows that