The Multifractal Random Walk as Pathwise Stochastic Integral: Construction and Simulation

Soledad Torres () and Ciprian A. Tudor ()
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Soledad Torres: Universidad de Valparaíso
Ciprian A. Tudor: Universidad de Valparaíso

Journal of Theoretical Probability, 2018, vol. 31, issue 1, 445-465

Abstract: Abstract 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.

Keywords: Malliavin calculus; Multifractal random walk; Pathwise integration; Scaling; Infinitely divisible cascades; Skorohod integral; 60C30; 60H07; 60H05 (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s10959-016-0713-5

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