  In this paper, we study ergodic backward stochastic differential equations
(EBSDEs for short), for which the underlying diffusion is assumed to be
multiplicative and of at most linear growth. The fact that the forward process
has an unbounded diffusion is balanced with an assumption of weak dissipativity
for its drift. Moreover, the forward equation is assumed to be non-degenerate.
Like in [HMR15], we show that the solution of a BSDE in finite horizon T
behaves basically as a linear function of T, with a shift depending on the
solution of the associated EBSDE, with an explicit rate of convergence.
Finally, we apply our results to an ergodic optimal control problem. In
particular, we show the large time behaviour of viscosity solution of
Hamilton-Jacobi-Bellman equation with an exponential rate of convergence when
the undelrying diffusion is multiplicative and unbounded.
