Higher-order adaptive methods for exit times of Itô diffusions
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We construct a higher-order adaptive method for strong approximations of exit times of Itô stochastic differential equations (SDE). 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 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 Itô–Taylor scheme of order 1.5 as numerical integrator and three step-sizes for adaptive time-stepping. For any ξ>0, we prove that the strong error is bounded by 𝒪(h^1-ξ) and 𝒪(h^3/2-ξ) for the first and second method, respectively. Under some conditions, we show that the expected computational cost of both methods are bounded by 𝒪(h^-1 |log(h)|), indicating that both methods are tractable. The theoretical results are supported by numerical examples, and we discuss the potential for extensions that improve the strong convergence rate even further.
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