Higher-order adaptive methods for exit times of Itô diffusions

Abstract

We construct a higher-order adaptive method for strong approximations of exit times of Itô stochastic differential equations (SDEs). The method employs a strong Itô–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 complement each other nicely: adaptive timestepping 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 the numerical integrator and two step sizes for adaptive timestepping: $h$ 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 Itô–Taylor scheme of order 1.5 as the numerical integrator and three step sizes for adaptive timestepping. Under some regularity assumptions, we show that for any $\xi>0$, the strong error is ${\mathcal{O}}(h^{1-\xi })$ and ${\mathcal{O}}(h^{3/2-\xi })$ for the first and second method, respectively. Provided quite restrictive commutativity conditions hold for the diffusion coefficient, we further show that the expected computational cost for both methods is ${\mathcal{O}}(h^{-1} \log (h^{-1}))$. This results in a near doubling/trebling of the strong error rate compared to the standard Euler–Maruyama-based approach, while the computational cost rate is kept close to order one. Numerical examples that support the theoretical results are provided, and we discuss the potential for extensions that would further improve the strong convergence rate of the method.