The multifractal random walk as pathwise stochastic integral: construction and simulation (Q1745273)

The authors define a multifractal random walk \(X(t)\) which formally represents the limit as the parameter \(r\to 0\), of the stochastic integral \[ X_r(t) = \int_0^t Q_r(u)dW(u). \] Here \(W\) is a Wiener process and \(Q\) is an infinitely divisible cascading noise, defined with respect to a given random measure. It is not assumed that \(Q\) is adapted to the filtration generated by \(W\), nor that \(W\) and \(Q\) are independent (a standard assumption in constructions of this type). Thus a different approach is needed to construct the stochastic integrals and to analyze their behavior as \(r\to 0\). The author's approach is to define a family of related stochastic integrals \[ X^+_r(t) = \int_0^t Q_r(u)d^+W^{(1)}(u) \] as the limit of a sequence of Riemann sums, and then to construct \(X(t)\) as the renormalized limit \[ X(t) = \lim_{r\to 0} \sqrt rX^+_r(t), \] where convergence is shown in the almost sure sense. The authors go on to show that the process \(X(t)\) satisfies an \(L^2\)-Lipschitz condition and the paper concludes with some numerical simulations.