# Continuous cascades in the wavelet space as models for synthetic turbulence

## TL;DR

This paper introduces a new class of stochastic processes based on wavelet transforms that model turbulence and financial market fluctuations, capturing multifractal and skewed properties with convergence guarantees.

## Contribution

It presents a novel wavelet-based multifractal cascade model that extends previous frameworks and accounts for asymmetry and leverage effects in turbulence and finance.

## Key findings

- Model exhibits stochastic self-similarity and multifractal scaling.
- Provides stationary, grid-free multifractal functions.
- Can reproduce skewness and leverage effects observed in real data.

## Abstract

We introduce a wide family of stochastic processes that are obtained as sums of self-similar localized "waveforms" with multiplicative intensity in the spirit of the Richardson cascade picture of turbulence. We establish the convergence and the minimum regularity of our construction. We show that its continuous wavelet transform is characterized by stochastic self-similarity and multifractal scaling properties. This model constitutes a stationary, "grid free", extension of $\cal W$-cascades introduced in the past by Arneodo, Bacry and Muzy using wavelet orthogonal basis. Moreover our approach generically provides multifractal random functions that are not invariant by time reversal and therefore is able to account for skewed multifractal models and for the so-called "leverage effect". In that respect, it can be well suited to providing synthetic turbulence models or to reproducing the main observed features of asset price fluctuations in financial markets.