Higher-order adaptive multilevel Monte Carlo methods for stochastic differential equations
Year: 2024
Type: dissertation
Abstract: Stochastic differential equations (SDEs) play an important role in modeling uncerctainty associated with the occurrence of real-world phenomena. For example, consider the field of computational finance where a lot of financial portforlio management companies rely on traders to make accurate predicitions regarding how the stock market would behave in the future. This enables them to make investments accordingly so as to maximize the profits from the fluctuations in the stock market. In the field of molecular chemistry, SDEs play a crucial role in modeling the random movement of molecules and atoms which thereby helps scientists predict long-term behaviors of the molecular systems. We refer the readers to the articles (Karatzas 1988, Azze 2024, Seydel 2017, Leimkuhler 2015) for further details regarding how stochastic differential equations are used to model uncertainties in financial and biological systems. In most cases, we are unable to compute the closed-form solution of SDEs. Even if we are able to obtain a closed-form solution of SDEs, it is oftentimes difficult to compute quantities of interest (QoIs) that depend on the measure associated with the given SDE. To compute weak approximations of QoIs, we generally solve the partial differential equation (PDE) associated with it, for example, the Fokker--Planck PDE or the Feynman--Kac PDE. However, solving the PDE is tractable neither analytically nor numerically as the dimensionality of the problem increases. To remedy this, we make use of numerical methods to approximate the solutions of SDEs, and use Monte Carlo methods to compute expected values of different QoIs. However, generating random variables needed for running Monte Carlo simulations is computationally expensive. Applying crude Monte Carlo method is also not great as the dimensionality of the problem increases. This especially holds true for the computation of exit times and ergodic measures of SDEs. To this end, utilizing the techniques that we develop in our thesis provides a better computational performance in comparison to the current state-of-the-art methods. Our fundamental idea is to develop higher-order numerical methods for approximating the solution of SDEs and to compute expected values of QoIs using the multilevel Monte Carlo (MLMC) method. This is an extension of the work done during my master's thesis.In the first part of our thesis, we develop a higher-order adaptive timestepping method for strong approximations of exit times of Itô diffusions. By using strong Itô--Taylor schemes for the numerical approximation of the SDE paths, alongside using finer timestep sizes as the numerical process gets closer to the boundary of the domain. The combination of higher-order numerical integration schemes and adaptive timestepping complement each other nicely in the following way: use of adaptive timestepping improves the accuracy of the exit time by reducing the magnitude of the overshoot of the numerical solution when it exits the domain and higher-order numerical integration schemes help improve the approximation of the state of the diffusion process. From numerical experiments, we observe that by using order $\gamma$ strong numerical integration schemes, for all $\gamma \geq 1$, alongwith a suitable adaptive timestepping strategy, we can improve the strong exit time error convergence. This in turn helps us achieve good computational cost savings when combined with the MLMC method. In the second part of the thesis, we develop a higher-order, spring-coupled, change-of-measure MLMC method for computing weak approximations of ergodic measures of SDEs with drift coefficients that do not satisfy the contractivity condition. By choosing a suitable spring term, the pairwise coupled trajectories of the MLMC method (employing the order 3/2 strong Itô--Taylor scheme) are pulled close to each other. This in turn helps us recover the contractivity property needed for the efficient implementation of the MLMC method. The numerical experiments indicate that the variance of the newly developed higher-order change-of-measure MLMC method grows linearly in time as opposed to using the standard MLMC method.
Source: RWTH Publications (RWTH Aachen)
Author: Ragunathan, Sankarasubramanian
Institution: RWTH Aachen University
Language: English
Subfield: Finance
Domain: Social Sciences
Open Access status: green
OpenAlex ID: W6963065095