Stability and numerical solutions of nonlinear implicit stochastic differential equations

This paper investigates the solvability, stability, and convergence of numerical solutions for nonlinear stochastic differential-algebraic equations (SDAEs) with time-varying coefficients. We establish sufficient conditions for solvability and stochastic asymptotic stability, and prove that the explicit Euler method converges as the mesh size tends to zero. Moreover, we show that the solvability conditions are also necessary in the linear case. Several examples are presented to illustrate the theoretical results. To the best of our knowledge, these results are new for time-varying SDAEs.