  We introduce tools for inference in the multifractal random walk introduced
by Bacry et al. (2001). These tools include formulas for smoothing, filtering
and volatility forecasting. In addition, we present methods for computing
conditional densities for one- and multi-step returns. The inference techniques
presented in this paper, including maximum likelihood estimation, are applied
to data from the Oslo Stock Exchange, and it is observed that the volatility
forecasts based on the multifractal random walk have a much richer structure
than the forecasts obtained from a basic stochastic volatility model.
