Pathwise Stability and Positivity of Semi-Discrete Approximations of the Solution of Nonlinear Stochastic Differential Equations

Ioannis S. Stamatiou ()
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Ioannis S. Stamatiou: University of West Attica

A chapter in Mathematical Analysis in Interdisciplinary Research, 2021, pp 859-873 from Springer

Abstract: Abstract We use the main idea of the semi-discrete method, originally proposed in (N. Halidias, International Journal of Computer Mathematics, 89(6) (2012), 780–794), to reproduce qualitative properties of a class of nonlinear stochastic differential equations with non-negative, non-globally Lipschitz coefficients and a unique equilibrium solution. The proposed fixed-time step method preserves the positivity of the solution and reproduces the almost sure asymptotic stability behavior of the equilibrium with no time-step restrictions. In particular, we are interested in the following class of scalar stochastic differential equations, x t = x 0 + ∫ 0 t x s a ( x s ) d s + ∫ 0 t x s b ( x s ) d W s , $$\displaystyle x_t =x_0 + \int _0^t x_sa(x_s)ds + \int _0^t x_sb(x_s)dW_s, $$ where a(⋅), b(⋅) are non-negative functions with b(u) ≠ 0 for u ≠ 0, x 0 ≥ 0 and {W t}t≥0 is a one-dimensional Wiener process adapted to the filtration { ℱ t } t ≥ 0 $$\{{\mathcal F}_t\}_{t\geq 0}$$ .

Date: 2021
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Persistent link: https://EconPapers.repec.org/RePEc:spr:spochp:978-3-030-84721-0_34

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DOI: 10.1007/978-3-030-84721-0_34

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