This thesis explores three different topics that relate the stochastic differential equations (SDE), including SDE with jumps, elliptic equations driven by Brownian motion with singular drift and reflected Brownian motion with singular drift. For the SDE of jump types, we consider stochastic differential equations driven by compensated Poisson random measure. We showed that the solution of the SDE admits non-explosion for a class of super-linear coefficients, and moreover, we proved that the pathwise uniqueness holds under certain non-Lipschitzian conditions on the coefficients. Moreover, we obtained the existence and uniqueness of solution u of elliptic equation associated with Brownian motion with singular drift. We then used the regularity of the weak solution u and the Zvonkin-type transformation to show that there is a unique weak solution to a stochastic differential equation when the drift is a measurable function. Furthermore, we showed that there exists a unique weak solution to the reflected Brownian motion with singular drift Î¼, where Î¼ is a vector-valued Kato class measure on Rd . To serve the purpose of monitoring the weak solution, we also established some Gaussian type estimates of the transition density function of the solution.
|Date of Award
|1 Aug 2021
- The University of Manchester
|Tusheng Zhang (Supervisor) & Denis Denisov (Supervisor)