Before revision:  The strength of our formula is that it only involves linear combinations of the exponential of the  generator matrix, as in the one-dimensional case. The framework developed allows then for easy-to-implement solutions when it comes to financial pricing, such as for path-dependent options or in a stochastic volatility models context.
After revision:  The strength of our formula is that it only involves linear combinations of the exponential of the generator matrix, extending the well-known moment formula for polynomial processes. The developed framework allows to replace costly simulations with more accurate estimates, and it may be used for increasing the accuracy in financial pricing, such as for path-dependent options or in a stochastic volatility models context.