| Abstract: |
| Consider a multi-dimensional stochastic differential equation with smooth coefficients. It is well-known that the Milstein scheme, with step size $h\in(0,1)$, provides an $O(h)$ pathwise approximation when the diffusion matrix satisfies the commutativity condition. In the general case, the same convergence rate was achieved by M. Wiktorsson, who studied the joint distribution of double stochastic integrals. An alternative approach, the coupling method, was devised by A. Davie, and he managed to approximate the solution up to any order when the diffusion matrix is nondegenerate. In the degenerate case he also found a close coupling for the double integrals, giving a genuine improvement to Wiktorsson`s result. However, Davie`s result does not work for longer iterated integrals. The work to be presented investigates the distribution of the \textit{triple} stochastic integral, and gives a coupling that leads to an $O(h^{3/2})$-approximation. The main idea is based on Davie`s coupling method and the Fourier representation by P. Kloeden and E. Platen. |
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