Weak Ito–Taylor schemes for geometric stochastic differential equations on the S ² manifold

This paper presents geometry-preserving weak solution techniques for Ito stochastic differential equations (SDEs) whose solutions are constrained to evolve on a geometrical space. Weak solutions are required where estimation of the average behaviour of the solution paths of an SDE is useful. The weak approximation to the SDE on the manifold is developed by exploiting the relation between the manifold and its Lie algebra through exponential mapping. The work demonstrates three weak integration schemes up to order 3.0 based on the Ito–Taylor expansion of the SDE on Lie algebra with associated stochastic integrals. Detailed formulation of the geometric operators associated with the higher-order schemes is shown along with convergence analysis. Effects of additive and multiplicative noise in dynamical systems on the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>S</mml:mtext> <mml:mn>2</mml:mn> </mml:msup> </mml:math> manifold are explored. The accuracy of the proposed integration schemes is shown to be consistent with the order of weak accuracy and it is shown that higher-order weak schemes provide better solution estimates and accurate description of the stable regions for the nonlinear dynamical systems on the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>S</mml:mtext> <mml:mn>2</mml:mn> </mml:msup> </mml:math> manifold even at coarser time steps. The proposed method accommodates a wide range of stochastic dynamical systems on the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>S</mml:mtext> <mml:mn>2</mml:mn> </mml:msup> </mml:math> manifold whose dynamics can be represented by an Ito SDE.