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Pathwise approximation of stochastic differential equations on domains: Higher order convergence rates without global Lipschitz coefficients. (English) Zbl 1163.65003
Summary: We study the approximation of stochastic differential equations on domains. For this, we introduce modified Itô-Taylor schemes, which preserve approximately the boundary domain of the equation under consideration. Assuming the existence of a unique non-exploding solution, we show that the modified Itô-Taylor scheme of order \(\gamma \) has pathwise convergence order \(\gamma - \varepsilon \) for arbitrary \(\varepsilon > 0\) as long as the coefficients of the equation are sufficiently differentiable. In particular, no global Lipschitz conditions for the coefficients and their derivatives are required. This applies for example to the so called square root diffusions.

65C30 Numerical solutions to stochastic differential and integral equations
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
65L20 Stability and convergence of numerical methods for ordinary differential equations
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