Browsing by Author "Sun, Fandi"

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Adaptive Milstein methods for stochastic differential equations
(Heriot-Watt University, 2023-02) Sun, Fandi; Lord, Professor Gabriel J.; Kelly, Doctor Conall
It was shown in [27] that the Euler-Maruyama (EM) method fails to converge with equidistant timesteps in the strong sense to the solutions of stochastic differential equations (SDEs) when either of the drift or diffusion coefficients is not globally Lipschitz continuous. Higher-order methods or schemes that are developed based on EM, e.g. Milstein method or EM with jumps, inherit the problem. We introduce an explicit adaptive Milstein method for SDEs with no commutativity condition. The drift and diffusion are separately locally Lipschitz and together satisfy a monotone condition. This method relies on a class of path-bounded time-stepping strategies which work by reducing the stepsize as solutions approach the boundary of a sphere, invoking a backstop method in the event that the timestep becomes too small. We prove that such schemes are strongly L2 convergent of order one. This order is inherited by an explicit adaptive EM scheme in the additive noise case. Moreover, we show that the probability of using the backstop method at any step can be made arbitrarily small. We compare our method to other fixed-step Milstein variants on a range of test problems. Secondly, we introduce a jump-adapted adaptive Milstein (JAAM) method for SDEs driven by Poisson random measure. With the conditions of drift and diffusion coefficients remaining the same as for the adaptive Milstein method, and the jump coefficient is globally Lipschitz continuous. The corresponding time-stepping strategies that we propose are hence path-bounded and also jump-adapted. We prove the L2 strong convergence of order one for JAAM and compare its computational efficiency with jump-adapted and fixed-step methods on test models.