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In ergodic stochastic problems the limit of the value function $V_\lambda$ ofthe associated discounted cost functional with infinite time horizon isstudied, when the discounted factor $\lambda$ tends to zero. These problemshave been well studied in the literature and the used assumptions guaranteethat the value function $\lambda V_\lambda$ converges uniformly to a constantas $\lambda\to 0$. The objective of this work consists in studying theseproblems under assumptions, namely, the nonexpansivity assumption, under whichthe limit function is not necessarily constant. Our discussion goes beyond thecase of the stochastic control problem with infinite time horizon and discussesalso $V_\lambda$ given by a Hamilton-Jacobi-Bellman equation of second orderwhich is not necessarily associated with a stochastic control problem. On theother hand, the stochastic control case generalizes considerably earlier worksby considering cost functionals defined through a backward stochasticdifferential equation with infinite time horizon and we give an explicitrepresentation formula for the limit of $\lambda V_\lambda$, as $\lambda\to 0$.

Representation of asymptotic values for nonexpansive stochastic control systems