Improved local approximation for multidimensional diffusions: The G-rates

We consider the problem of improving the local approximations for multidimensional diffusions. In particular, our proposed explicit approximation improves the Milshtein approximation. We also provide a semi-explicit convergence rate estimate (we call it G-rate) for the proposed local approximation. The main error term in the difference of densities is bounded by a polynomial multiplied by a Gaussian density and the remainder is exponentially small as time goes to zero.