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 \(r\rightarrow 0\) of the family of stochastic processes \((X_{r})_{r>0}\) where
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 \(X_{r}(t)\) 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.
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Soledad Torres supported by the Grant MEC-FEDER Ref. MTM2006-06427, FONDECYT 1130586, Red de Análisis Estocástico y Aplicaciones ACT 1112.
Both authors are supported by the Mathamsud 16-MATH-03 SIDRE project and by the ECOS program C15E05.
Appendix: Elements of Malliavin Calculus
Appendix: Elements of Malliavin Calculus
Let \((W_{t})_{t\in T}\) be a classical Wiener process on a standard Wiener space \(\left( \Omega ,{\mathcal {F}},\mathbf {P}\right) \). By \(W(\varphi )\), we denote the Wiener integral of the function \(\varphi \in L^{2}(T, m)\) with respect to the Brownian motion W (the measure m is given by (12) in our paper). We denote by D the Malliavin derivative operator that acts on smooth functionals of the form \(F=g(W(\varphi _{1}), \ldots , W(\varphi _{n}))\) (here g is a smooth function with compact support and \(\varphi _{i} \in L^{2}(T)\) for \(i=1,..,n\))
The operator D can be extended to the closure \(\mathbb {D}^{p,2}\) of smooth functionals with respect to the norm
where the i th Malliavin derivative \(D^{(i)}\) is defined iteratively. The adjoint of D is denoted by \(\delta \) and is called the divergence (or Skorohod) integral. Its domain (\(Dom(\delta )\)) coincides with the class of stochastic processes \(u\in L^{2}(\Omega \times T)\) such that
for all \(F\in \mathbb {D}^{1,2}\) and \(\delta (u)\) is the element of \(L^{2}(\Omega )\) characterized by the duality relationship
For adapted integrands, the divergence integral coincides with the classical Itô integral. A subset of \(Dom (\delta )\) is the space \(\mathbb {L} ^{1,p}\) of the stochastic processes such that \(u_{t}\) is Malliavin differentiable for every t and
We will need Meyer’s inequality that allows to estimate the \(L^{p} \) moment of the Skorohod integral
For the \(L^{2}\) moment of the Skorohod integral we have the explicit formula
if \(u\in \mathbb {L} ^{1,2}\). We also recall that the Malliavin derivative satisfies the chain rule
if f is a differentiable function and \(F\in \mathbb {D}^{1,2}\). A last formula we need is the integration by parts formula
if \(F\in \mathbb {D} ^{1,2}, u\in Dom (\delta ).\)
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Torres, S., Tudor, C.A. The Multifractal Random Walk as Pathwise Stochastic Integral: Construction and Simulation . J Theor Probab 31, 445–465 (2018). https://doi.org/10.1007/s10959-016-0713-5
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DOI: https://doi.org/10.1007/s10959-016-0713-5
Keywords
- Malliavin calculus
- Multifractal random walk
- Pathwise integration
- Scaling
- Infinitely divisible cascades
- Skorohod integral