Quaternion version of the Itô’s formula

The classic Itô’s formulas in both real and complex fields are widely used to cope with the stability analysis of various stochastic systems. In this paper, the quaternion version of the Itô’s formula is established by resorting to the properties and the operation rules of quaternion, which covers the real and complex versions as special cases and can be applied to the stability analysis of stochastic quaternion-valued systems. Two examples are provided for demonstrating the validness of the proposed formula.