This talk is devoted to the study of the large time behaviour of  solution of some semilinear parabolic partial differential equation (with Dirichlet or Neumann boundary condition). A probabilistic method (more precisely, an approach via an ergodic backward stochastic equation) is developped to show that the solution of a parabolic semilinear PDE at large time  $T$ behaves like a linear term $/lambda T$ shifted  with a function $v$, where $(v,/lambda)$  is the solution of the ergodic PDE associated to the parabolic PDE.  The advantage of our method is that it gives an explicit rate of convergence. The result gives a perspective to give a precise estimate on the long run asymptotics for utility maximisation.

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