The Multifractal Random Walk As Pathwise Stochastic Integral: Construction And Simulation

We define a multifractal random walk (MRW) as an anticipating pathwise integral, as limit of Riemann sums. The MRW is usually defined as the limit as r→ 0 of the family of stochastic processes (Xr)rCloseSPigtSPi0 where (Formula presented.) Xr(t)=∫0tQr(u)dW(u),t≥0,where W is a Wiener process and Q an infinitely divisible cascading noise (IDC noise) not adapted to the filtration generated by W. In order to define the stochastic integral Xr(t) and to simulate it, one usually assumes that Q and W are independent. Our purpose is to define the MRW with a dependence structure between the IDC noise Q and the Wiener process W. Our construction is done by using Riemann sums, and it allows the simulation of the process.