E-Thesis 48 views 19 downloads
Convergence of Numerical Solutions of Stochastic Differential Delay Equations / ULISES MUNOZ
Swansea University Author: ULISES MUNOZ
DOI (Published version): 10.23889/SUThesis.67952
Abstract
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...
| Published: |
Swansea University, Wales, UK
2024
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| Institution: | Swansea University |
| Degree level: | Doctoral |
| Degree name: | Ph.D |
| Supervisor: | Yuan, C. |
| URI: | https://cronfa.swan.ac.uk/Record/cronfa67952 |
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| Abstract: |
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. |
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| Item Description: |
A selection of content is redacted or is partially redacted from this thesis to protect sensitive and personal information. |
| Keywords: |
Adaptive Euler Maruyama scheme; McKean-Vlasov Stochastic differential delay equations (MV-SDDEs); Strong convergence; Boundedness of the pth- moments; Almost sure exponential stability; Multilevel Monte Carlo simulation; Variance of two coupled paths |
| College: |
Faculty of Science and Engineering |