  We study the point of transition between complete and incomplete financial
models thanks to Dirichlet Forms methods. We apply recent techniques,
developped by Bouleau, to hedging procedures in order to perturbate parameters
and stochastic processes, in the case of a volatility parameter fixed but
uncertain for traders; we call this model Perturbed Black Scholes (PBS) Model.
We show that this model can reproduce at the same time a smile effect and a
bid-ask spread; we exhibit the volatility function associated to the
local-volatility model equivalent to PBS model when vanilla options are
concerned.
  Lastly, we present a connection between Error Theory using Dirichlet Forms
and Utility Function Theory.
