A note on pathwise stability and positivity of nonlinear stochastic differential equations
For researchers studying stochastic differential equations, this provides a numerical method that preserves key qualitative properties without restrictive conditions.
The paper uses a semi-discrete method to preserve positivity and almost sure asymptotic stability of nonlinear stochastic differential equations with nonnegative coefficients, achieving stability without time-step restrictions.
We use the semi-discrete method, originally proposed in Halidias (2012), Semi-discrete approximations for stochastic differential equations and applications, International Journal of Computer Mathematics, 89(6), to reproduce qualitative properties of a class of nonlinear stochastic differential equations with nonnegative, non-globally Lipschitz coefficients and a unique equilibrium solution. The proposed fixed-time step method preserves the positivity of solutions and reproduces the almost sure asymptotic stability behavior of the equilibrium with no time-step restrictions.