The multidimensional Log S-fBM model (m-Log S-fBM) extends the Log S-fBM model proposed by Wu \textit{et al.} to the multidimensional setting. We first define the multidimensional Stationary fractional Brownian motion process (m-S-fBM), whose marginals are S-fBM processes characterized by the correlation limit $T$, the intermittency parameters $\{\lambda_i^2\}_{1\leq i \leq d}$, and the Hurst exponents $\{H_i\}_{1\leq i \leq d}$. The cross-dependence structure is captured by the co-intermittency matrix $\{\xi_{i,j}\}_{1\leq i,j \leq d}$ and the co-Hurst matrix $\{H_{i,j}\}_{1\leq i,j \leq d}$.

Subsequently, we introduce the m-Log S-fBM, in which each marginal follows a Log S-fBM process, while their mutual dependencies are governed by the covariance structure of the underlying m-S-fBM. Like the one dimensional Log S-fBM model, we demonstrate that the mLog S-fBM is well defined for coHurst entries in $\left]0,\frac{1}{2}\right[^{d \times d}$ as well as for a vanishing coHurst matrix enabling to bridge the rough and the multifractal regimes. Within this framework, we extend the small-intermittency approximation from the univariate to the multivariate case, allowing the approximation of generalized moments of log-multifractal random measures by those of the integrated S-fBM process. This enables the estimation of cross-covariation parameters between any pair of marginals $i$ and $j$ via the Generalized Method of Moments (GMM), whose accuracy we validate on synthetically generated data.

Finally, we apply this formalism to financial modeling by constructing a multidimensional stochastic volatility model, calibrated using S\&P 500 market data. Empirical results show that the off-diagonal entries of the co-Hurst matrix are close to the Hurst exponent of the S\&P 500 itself, while the off-diagonal entries of the co-intermittency matrix are consistent with its intermittency parameters.