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In this paper, we study ergodic backward stochastic differential equations(EBSDEs for short), for which the underlying diffusion is assumed to bemultiplicative and of at most linear growth. The fact that the forward processhas an unbounded diffusion is balanced with an assumption of weak dissipativityfor 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 Tbehaves basically as a linear function of T, with a shift depending on thesolution of the associated EBSDE, with an explicit rate of convergence.Finally, we apply our results to an ergodic optimal control problem. Inparticular, we show the large time behaviour of viscosity solution ofHamilton-Jacobi-Bellman equation with an exponential rate of convergence whenthe undelrying diffusion is multiplicative and unbounded.

Ergodic BSDE with an unbounded and multiplicative underlying diffusion and application to large time behavior of viscosity solution of HJB equation