To extend the skew-symmetric scheme to handle correlated volatility structures, one could consider the following approaches:
Generalized Volatility Models: Instead of assuming diagonal volatility, one could model the volatility matrix as a function of the state variable that captures correlations between different dimensions. This would involve defining a volatility function σ(x) that is a full matrix rather than a diagonal one, allowing for interactions between different stochastic processes.
Cholesky Decomposition: One practical method to incorporate correlated volatilities is to use the Cholesky decomposition of the volatility matrix. By expressing the volatility matrix as σ(x) = L(x)L(x)ᵀ, where L(x) is a lower triangular matrix, one can generate correlated random variables by transforming independent standard normal variables through this decomposition.
Multivariate Skew-Symmetric Distributions: The proposal generating mechanism could be adapted to use multivariate skew-symmetric distributions, which would allow for the introduction of correlation structures directly into the sampling process. This would involve defining the probability functions pi in a way that accounts for the joint distribution of the increments.
Adaptive Step-Size Control: Implementing an adaptive step-size control mechanism that adjusts the step size based on the estimated volatility could help manage the complexities introduced by correlated volatilities. This would ensure that the numerical scheme remains stable and convergent even in the presence of strong correlations.