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 $$r\rightarrow 0$$ r → 0 of the family of stochastic processes $$(X_{r})_{r>0}$$ ( X r ) r > 0 where $$\begin{aligned} X_{r}(t)=\int _{0} ^ {t} Q_{r}(u)\hbox {d}W(u), \quad t\ge 0, \end{aligned}$$ X r ( t ) = ∫ 0 t Q r ( u ) d W ( 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 $$X_{r}(t)$$ X r ( 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.