\[ \begin{array} { l } { { \displaystyle Y _ { T } = \operatorname* { l i m } _ { n \to \infty } \left( v ^ { \alpha _ { n } } ( X _ { T } ) - v ^ { \alpha _ { n } } ( x _ { 0 } ) \right) } } \\ { { \displaystyle \quad = \operatorname* { l i m } _ { n \to \infty } \left( v ^ { \alpha _ { n } } ( X _ { t } ) - v ^ { \alpha _ { n } } ( x _ { 0 } ) \right) - \operatorname* { l i m } _ { n \to \infty } \displaystyle \sum _ { t \leq u < T } \left( f ( X _ { u } , Z _ { u } ^ { \alpha _ { n } } ) - \alpha _ { n } v ^ { \alpha _ { n } } ( X _ { u } ) \right) } } \\ { { \displaystyle \qquad + \operatorname* { l i m } _ { n \to \infty } \displaystyle \sum _ { t \leq u < T } \left( Z _ { u } ^ { \alpha _ { n } } \right) ^ { * } M _ { u + 1 } } } \\ { { \displaystyle \qquad = v ( X _ { t } ) - \operatorname* { l i m } _ { n \to \infty } \displaystyle \sum _ { t \leq u < T } \left( f ( X _ { u } , Z _ { u } ^ { \alpha _ { n } } - v ^ { \alpha _ { n } } ( x _ { 0 } ) \mathbf { 1 } ) - \alpha _ { n } v ^ { \alpha _ { n } } ( X _ { u } ) \right) } } \\ { { \displaystyle \qquad + \operatorname* { l i m } _ { n \to \infty } \displaystyle \sum _ { t \leq u < T } \left( Z _ { u } ^ { \alpha _ { n } } - v ^ { \alpha _ { n } } ( x _ { 0 } ) \mathbf { 1 } \right) ^ { * } M _ { u + 1 } } } \\ { { \displaystyle \qquad = Y _ { t } - \displaystyle \sum _ { t \leq u < T } \left( f ( X _ { u } , Z _ { u } ) - \lambda \right) + \displaystyle \sum _ { t \leq u < T } Z _ { u } ^ { * } M _ { u + 1 } , } } \end{array} \]

and we see that \( ( Y , Z , \lambda ) \) is indeed a solution of the EBSDE (4.4).

Suppose that \( ( Y ^ { \prime } , Z ^ { \prime } , \lambda ^ { \prime } ) \) is another bounded solution. Let \( \tilde { Y } \, = \, Y \, - \, Y ^ { \prime } \),\( \tilde { Z } = Z - Z ^ { \prime } \) and \( \tilde { \lambda } = \lambda - \lambda ^ { \prime } \). Then

\[ \tilde { Y } _ { T } = \tilde { Y } _ { 0 } - \sum _ { 0 \leq u < T } \left( f ( X _ { u } , Z _ { u } ) - f ( X _ { u } , Z _ { u } ^ { \prime } ) - \tilde { \lambda } \right) + \sum _ { 0 \leq u < T } \tilde { Z } _ { u } ^ { \ast } M _ { u + 1 } . \qquad ( 4 . 5 ) \]

By Proposition 2.2, there exists a measure Q such that

\[ - \sum _ { 0 \leq u < t } \left( f ( X _ { u } , Z _ { u } ) - f ( X _ { u } , Z _ { u } ^ { \prime } ) \right) + \sum _ { 0 \leq u < t } \tilde { Z } _ { u } ^ { \ast } M _ { u + 1 } \]

is a martingale under Q. Taking an \( \mathbb { E } _ { \mathbb { Q } } \) expectation in (4.5), we obtain

\[ \begin{array} { r } { \tilde { \lambda } = T ^ { - 1 } \mathbb { E } _ { \mathbb { Q } } \big [ \tilde { Y } _ { T } - \tilde { Y } _ { 0 } \big ] . } \end{array} \]

Since \( \tilde { Y } \) is uniformly bounded, taking the limit \( T \to \infty \) gives \( \tilde { \lambda } \) = 0, so that \( \lambda = \lambda ^ { \prime } \). Substituting back into (4.5) and taking an \( \mathbb { E } _ { \mathbb { O } } [ \, \cdot \, | \, \mathcal { F } _ { 0 } ] \)] expectation gives

\[ \mathbb { E } _ { \mathbb { Q } } { \left[ \tilde { Y } _ { T } \mid \mathcal { F } _ { 0 } \right] } = \tilde { Y } _ { 0 } . \eqno ( 4 . 6 ) \]

Suppose further that \( Y ^ { \prime } \) and \( Z ^ { \prime } \) are Markovian and do not depend on t, so that in particular there exists a function \( v ^ { \prime } : \mathcal { X } \rightarrow \mathbb { R } \) such that \( Y _ { t } ^ { \prime } \, = \, v ^ { \prime } ( X _ { t } ) \). Then the measure Q may be taken to be the measure given by Lemma 4.1, so that X is still a uniformly ergodic Markov chain under Q. Writing \( \tilde { \pi } \) for the ergodic measure of X under Q, it follows from (4.6) that, for any \( x \in \mathcal { X } \),

\[ v ( x ) - v ^ { \prime } ( x ) = \operatorname* { l i m } _ { T \to \infty } \mathbb { E } _ { \mathbb { Q } } \big [ \tilde { Y } _ { T } \, \big | \, X _ { 0 } = x \big ] = \int _ { \mathcal { X } } \big ( v ( y ) - v ^ { \prime } ( y ) \big ) d \tilde { \pi } ( y ) . \]

Since the right hand side is independent of x, we see that \( v ( x ) = v ^ { \prime } ( x ) + c \) for all x, and hence that \( Y _ { t } = Y _ { t } ^ { \prime } + c \), for some \( c \in \mathbb { R } \). In particular, if \( v ^ { \prime } ( x _ { 0 } ) = 0 \), then \( c \) = 0, and hence \( Y = Y ^ { \prime } \) up to indistinguishability.

With \( \tilde { \lambda } \) = 0 and \( \tilde { Y } _ { t } = c = \tilde { Y } _ { t + 1 } \), we deduce from the one-step dynamics of \( \tilde { Y } \) that \( \tilde { Z } _ { t } ^ { * } M _ { t + 1 } = 0 , \) and hence that \( Z \sim _ { M } Z ^ { \prime } \).