Numerical Studies for Petrol and Gas Reservoir Problems

The focus of this project will be mathematical models to describe petroleum or gas reservoir simulations. For such a kind of problems, permeability is desperately needed in the oil industry, however, it is hard to accurately measure the permeability field in the earth, due to the large area of oil reservoir and complicated earth structure. Therefore, identification of permeability parameter is crucial for the efficiency of the mathematical models. Since the classical (deterministic) partial differential equations cannot express the behavior of the physical problem completely, the permeability parameter is represented as a random field.  The mathematical modelling of such a kind of problems corresponds to basically unsteady convection diffusion equations. Therefore, our first interest will be numerical investigation of the time-dependent convection diffusion equation with random input data. Then, we extend the concept to the stochastic optimal control problems. Lastly, by constructing the random data parameters as permeability with the help of the original data, the effect of the mathematical simulations can be observed.

Backward Stochastic Differential Equations with Singular Terminal Values and Applications in Finance

This project  focuses on this type of BSDE where the terminal condition is allowed to depend on the entire path of the underlying process; we call such terminal conditions ''Non-Markovian.'' The goal is to construct solutions for a range of non-Markovian singular terminal values under various assumptions, and to understand, to the best of our ability, to what extent such BSDE can be solved. We will also explore possible applications of our results to finance. In a financial context, non-Markovian terminal conditions correspond to specifying conditions under which the liquidation takes place.