Forecasting multifractal volatility

This paper develops analytical methods to forecast the distribution of future returns for a new continuous-time process, the Poisson multifractal. The process captures the thick tails, volatility persistence, and moment scaling exhibited by many financial time series. It can be interpreted as a stochastic volatility model with multiple frequencies and a Markov latent state. We assume for simplicity that the forecaster knows the true generating process with certainty but only observes past returns. The challenge in this environment is long memory and the corresponding infinite dimension of the state space. We introduce a discretized version of the model that has a finite state space and an analytical solution to the conditioning problem. As the grid step size goes to zero, the discretized model weakly converges to the continuous-time process, implying the consistency of the density forecasts.