Adaptive timestepping for SDEs with non-globally Lipschitz drift

Thesis

In this thesis, we focus on the numerical approximation of SDEs with a drift which is not globally Lipschitz, and corresponding sensitivity calculations.

First, we propose an adaptive timestep construction for an Euler- Maruyama approximation of these SDEs over a finite time interval. It is proved that if the timestep is bounded appropriately, then over a finite time interval the numerical approximation is stable, and the expected number of timesteps is finite. Moreover, we extend this scheme to ergodic SDEs with contractivity over an infinite time interval and prove that the bound for moments and the strong error of the numerical solution are uniform in T, which allow us to introduce the adaptive multilevel Monte Carlo (MLMC) method to further improve the efficiency.

Next, we apply a new MLMC method for the ergodic SDEs without contractivity. By introducing a change of measure technique, we simulate the paths with contractivity and add the Radon-Nikodym derivative to the estimator. It is shown that the variance of the new level estimators increases linearly in T, which is a great reduction compared with the exponential increase in the standard MLMC.

Lastly, we derive a new pathwise sensitivity estimator for chaotic SDEs by introducing a spring term between the original and perturbed SDEs. The variance of the new estimator increases only linearly in time T; compared with the exponential increase of the standard pathwise estimator. We also consider using this estimator for the SDE with Richardson-Romberg extrapolation on the volatility parameter to approximate the sensitivities of the invariant measure of chaotic ODEs.

All of the analysis is supported by numerical results.

Authors: Fang, W

• DOI: 10.5287/ora-qo5rw7yy2
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Adaptive; Numerical method; Chaotic System; Ergodic SDE; Multilevel Monte Carlo
• Subjects: Mathematics