Strong convergence of a semi tamed scheme for stochastic differential algebraic equation under non-global Lipschitz coefficients

Abstract: We are investigating the first strong convergence analysis of a numerical method for stochastic differential algebraic equations (SDAEs) under a non-global Lipschitz setting. It is well known that the explicit Euler scheme fails to converge strongly to the exact solution of a stochastic differential equation (SDEs) when at least one of the coefficients grows superlinearly. The problem becomes more challenging in the case of stochastic differential-algebraic equations (SDAEs) due to the singularity of the matrix. To address this, we build a new scheme called the semi-implicit tamed method for SDAEs and provide its strong convergence result under non-global Lipschitz setting. In other words, the linear component of the drift term is approximated implicitly, whereas its nonlinear component is tamed and approximated explicitly. We show that this method strongly converges with order $\frac{1}{2}$ to the exact solution. To prove this strong convergence result, we first derive an equivalent scheme, that we call the dual tamed scheme, which is more suitable for mathematical analysis and is associated with the inherent stochastic differential equation obtained by eliminating the constraints from the original SDAEs. To demonstrate the effectiveness of the proposed scheme, numerical simulations are performed, confirming that the theoretical findings are consistent with the numerical results.