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Given that Donoho-Johnstone had studied their wavelet shrinkage denoising approach for Besov and Treibel spaces and we want to apply their results for long memory in stochastic volatility, it is useful to consider what would happen if the the model were fractional Brownian motion with Hurst exponent $H$.  In this case, David Nualart has shown more generally that the stochastic integral $\int_0^t u_s dW_s^H$ is in a Besov space.  In case $u_s \in L^{\delta,1}$ for some large $\delta>0$ then the Besov space is $B^H_{\infty,\infty}$.  Here is the paper:  NualartBesovRegularityFBM.  This is extremely convenient because it brings us closer to a rigorous application of the Donoho-Johnstone denoising theory where they had considered minimax for Besov spaces.  In actual application to long memory stochastic volatility one uses an ARFIMA model rather than fractional Brownian motion but one suspects that the pathwise regularity has similar properties in terms of the Hurst exponent $H=d+1/2$.