An Explicit Positivity-Preserving Method for Nonlinear Aït-Sahalia Model Driven by Fractional Brownian Motion

This paper develops an explicit positivity-preserving method for the nonlinear Aït-Sahalia interest rate model driven by fractional Brownian motion. To overcome the difficulties in obtaining the convergence rate of this positivity-preserving method, the Lamperti transformation is utilized, which gives an auxiliary equation. And the convergence rate of the numerical method for this auxiliary equation is obtained by virtue of Malliavin calculus. Naturally, the target follows from the inverse of the Lamperti transformation. As a byproduct, the convergence rate of the explicit positivity-preserving method for stochastic differential equations driven by fractional Brownian motion with symmetric coefficients is obtained. Finally, several numerical experiments are performed to verify the theoretical results and demonstrate the advantage of the explicit method.