Stochastic Differential Equations and Strict Local Martingales

In this thesis, we address two problems arising from the application of stochastic differential equations (SDEs). The first one pertains to the detection of asset bubbles, where the price process solves an SDE. We combine the strict local martingale model together with a statistical tool to instantaneously check the existence and severity of asset bubbles through the asset’s historical price process. Our approach assumes that the price process of interest is a CEV process. We relate the exponent parameter in the CEV process to an asset bubble by studying the future expectation and the running maximum of the CEV process. The detection of asset bubbles then boils down to the estimation of the exponent. With a dynamic linear regression model, inference on the exponent can be carried out using historical price data. Estimation of the volatility and calibration of the parameters in the dynamic linear regression model are also studied. When using SDEs in practice, for example, in the detection of asset bubbles, one often would like to simulate its paths using the Euler scheme to study the behavior of the solution. The second part of this thesis focuses on the convergence property of the Euler scheme under the assumption that the coefficients of the SDE are locally Lipschitz and that the solution has no finite explosion. We prove that if a numerical scheme converges uniformly on any compact time set (UCP) in probability with a certain rate under the globally Lipschitz condition, then when the globally Lipschitz condition is replaced with a locally Lipschitz one plus a no finite explosion condition, UCP convergence with the same rate holds. One contribution of this thesis is the proof of √n-weak convergence of the asymptotic normalized error process. The limit error process is also provided. We further study the boundedness for the second moment of the weak limit process and its running maximum under both the globally Lipschitz and the locally Lipschitz conditions. The convergence of the Euler scheme in the sense of approximating expectations of functionals is also studied under the locally Lipschitz condition.