Numerical approximation of stochastic differential equations with non-globally Lipschitz continuous coefficients

• Applicant: Professor Dr. Martin Hutzenthaler
• Subject Area: Mathematics
• Term: from 2012 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 219293315

• Final Report Year: 2018

Final Report Abstract

We developed a thorough theory for one-step methods which are recursively defined as a general function of the previous state, of the time increment and of the increment of the Brownian motion. For consistent one-step methods we established strong convergence for most of the stochastic ordinary differential equations (SODEs) with locally Lipschitz continuous coefficients having finite moments. A positive rate of convergence, however, does not exist in general. We have found SODEs with smooth and bounded coefficients whose solution is approximated by the Euler approximations in the strong sense but not with a positive rate. The associated semigroup of our counterexample has even the ’loss of regularity’ property that there exists a smooth function with compact support whose image under the semigroup is not locally Hölder continuous. We also established a positive convergence rate for a large class of SODEs. For this we assumed that the local Lipschitz constants of the coefficient functions are bounded by sums of certain functions and that both the exact solution and the numerical approximation have suitable (uniformly bounded) exponential moments. Finally we proved that these assumptions are satisfied for a large class of SODEs with non-globally monotone coefficients. Among these are the stochastic Duffing-van der Pol oscillator, the stochastic Lorenz equation with bounded noise and the Langevin dynamics.

Publications

• Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations. (2013), 1–54
  Cox, S. G., Hutzenthaler, M., and Jentzen, A.
  
• On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with non-globally monotone coefficients. (2014), 1–41
  Hutzenthaler, M., and Jentzen, A.
  
• Strong convergence rates and temporal regularity for Cox-Ingersoll-Ross processes and Bessel processes with accessible boundaries. (2014), 32 pages
  Hutzenthaler, M., Jentzen, A., and Noll, M.
  
• Loss of regularity for Kolmogorov equations. Ann. Probab. 43, 2 (2015), 468–527
  Hairer, M., Hutzenthaler, M., and Jentzen, A.
  
• Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients. Mem. Amer. Math. Soc. 4 (2015), 1–112
  Hutzenthaler, M., and Jentzen, A.
  
• Convergence in Hölder norms with applications to Monte Carlo methods in infinite dimensions. (2016), 38 pages
  Cox, S., Hutzenthaler, M., Jentzen, A., van Neerven, J., and Welti, T.
  
• Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations. Math. Comp. 87, 311 (2018), 1353–1413
  Hutzenthaler, M., Jentzen, A., and Wang, X.