Multifractal Fractional Ornstein-Uhlenbeck Processes

Abstract

The Ornstein-Uhlenbeck process can be seen as a paradigm of a finite-variance and statistically stationary rough random walk. Furthermore, it is defined as the unique solution of a Markovian stochastic dynamics and shares the same local regularity as the one of the Brownian motion. A natural generalization of this process able to reproduce the local regularity of a fractional Brownian motion of parameter H is provided by the fractional Ornstein-Uhlenbeck process. Based on previous works, we propose to include some Multifractal corrections to this picture using a Gaussian Multiplicative Chaos. The aforementioned process, called a Multifractal fractional Ornstein-Uhlenbeck process, is a statistically stationary finite-variance process. Its underlying dynamics is non-Markovian, although non-anticipating and causal. The numerical scheme and theoretical approach are based on a regularization procedure, that gives a meaning to this dynamical evolution, which unique solution converges towards a well-behaved stochastic process.