Adaptive timestepping for pathwise stability and positivity of strongly discretised nonlinear stochastic differential equations
This work addresses the challenge of preserving dynamical properties (stability and positivity) in numerical simulations of nonlinear SDEs, which is important for applications in finance, biology, and physics.
The authors developed an adaptive timestepping strategy for the Euler-Maruyama discretisation of nonlinear SDEs with non-negative, non-globally Lipschitz coefficients, achieving a.s. asymptotic stability and instability of the equilibrium and ensuring positivity with arbitrarily high probability.
We consider the use of adaptive timestepping to allow a strong explicit Euler-Maruyama discretisation to reproduce dynamical properties of a class of nonlinear stochastic differential equations with a unique equilibrium solution and non-negative, non-globally Lipschitz coefficients. Solutions of such equations may display a tendency towards explosive growth, countered by a sufficiently intense and nonlinear diffusion. We construct an adaptive timestepping strategy which closely reproduces the a.s. asymptotic stability and instability of the equilibrium, and which can ensure the positivity of solutions with arbitrarily high probability. Our analysis adapts the derivation of a discrete form of the Itô formula from Appleby et al (2009) in order to deal with the lack of independence of the Wiener increments introduced by the adaptivity of the mesh. We also use results on the convergence of certain martingales and semi-martingales which influence the construction of our adaptive timestepping scheme in a way proposed by Liu & Mao (2017).