Adaptive timestepping for pathwise stability and positivity of strongly discretised nonlinear stochastic differential equations
| Autor: | Cónall Kelly, Eeva Maria Rapoo, Alexandra Rodkina |
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| Rok vydání: | 2017 |
| Předmět: |
Discretization
Applied Mathematics 37H10 39A50 60H35 65C30 Mathematical analysis 010103 numerical & computational mathematics Numerical Analysis (math.NA) Lipschitz continuity 01 natural sciences Euler–Maruyama method Stability (probability) Instability 010101 applied mathematics Computational Mathematics Exponential stability Convergence (routing) FOS: Mathematics Mathematics - Numerical Analysis 0101 mathematics Independence (probability theory) Mathematics |
| DOI: | 10.48550/arxiv.1706.03098 |
| Popis: | 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 It�� 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). 24 pages, 2 figures |
| Databáze: | OpenAIRE |
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