Convergence of Numerical Solutions of Stochastic Differential Delay Equations - ULISES MUNOZ (2024)
In this thesis we investigate explicit numerical approximations for stochastic diﬀerential delay equations (SDDEs) under a local Lipschitz condition by employing the adaptive Euler-Maruyama (EM) method. Working in both ﬁnite and inﬁnite horizons, we achieve strong convergence results of the adaptive EM solution. We also obtain the order of convergence in ﬁnite horizon. In addition, we show almost sure exponential stability of the adaptive approximate solution for both SDEs and SDDEs. Further, we prove strong convergence of the adaptive solution for McKean-Vlasov SDDEs (MV-SDDEs). In the second part of the thesis, we estimate the variance of two coupled paths derived with the Multilevel Monte Carlo method combined with the EM discretization scheme for the simulation of MV-SDEs with small noise ﬁrst and for MV-SDDEs later. The result often translates into a more eﬃcient method than the standard Monte Carlo method combined with algorithms tailored to the small noise setting.