We construct a higher-order adaptive method for strong approximations of exit times of It\^o stochastic differential equations (SDE). The method employs a strong It\^o--Taylor scheme for simulating SDE paths, and adaptively decreases the step-size in the numerical integration as the solution approaches the boundary of the domain. These techniques turn out to complement each other nicely: adaptive time-stepping improves the accuracy of the exit time by reducing the magnitude of the overshoot of the numerical solution when it exits the domain, and higher-order schemes improve the approximation of the state of the diffusion process. We present two versions of the higher-order adaptive method. The first one uses the Milstein scheme as numerical integrator and two step-sizes for adaptive time-stepping: $h$ when far away from the boundary and $h^2$ when close to the boundary. The second method is an extension of the first one using the strong It\^o--Taylor scheme of order 1.5 as numerical integrator and three step-sizes for adaptive time-stepping. For any $\xi>0$, we prove that the strong error is bounded by $\mathcal{O}(h^{1-\xi})$ and $\mathcal{O}(h^{3/2-\xi})$ for the first and second method, respectively, and the expected computational cost for both methods is $\mathcal{O}(h^{-1} \log(h^{-1}))$. Theoretical results are supported by numerical examples, and we discuss the potential for extensions that improve the strong convergence rate even further.
翻译:我们为 It ⁇ o stochacistic 方程式( SDE) 的退出时间的强烈近似值构建了更高层次的适应方法。 这种方法使用强大的 It ⁇ o- Taylor 方案模拟 SDE 路径, 并在解决方案接近域界时, 适应性地降低数字整合的步数大小。 这些技术最终可以很好地互相补充: 适应性时间步骤可以提高退出时间的准确性, 当它退出域域时, 高层次方案会减少数字解决方案的超标范围, 并且改善扩散进程状态的近似值。 我们展示了两种更高层次适应方法的两种版本 。 对于任何 $\x\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\