GPU Computing and Machine Learning for solving high-dimensional BSDEs
My name is Lorenc Kapllani, I come from Albania. Currently, I’m following my first year of the PhD program in University of Wuppertal, group of Applied Mathematics and Numerical Analysis (AMNA). I completed the bachelor’s degree in Engineering Mathematics at the Polytechnic University of Tirana (Albania). My bachelor thesis was about Numerical Analysis and Simulation of Problems in Electrostatics and Fluid Flow. After researching in this area, I decided to follow my studies abroad in the same field. The master of science Computer Simulation in Science at University of Wuppertal triggered my attention. Therefore, I applied and was enrolled in this course, with Financial Mathematics as specialization. During the first year of my master studies, I also worked as a junior credit risk analyst in GEFA Bank GmbH. I wanted to get some practical background parallel with academic studies. Moreover, during my master thesis, I worked with the Numerical Solution of Backward Stochastic Differential Equations (BSDE) on GPU computing, considering applications in option pricing. After graduating, I worked as a Credit Risk Analyst in GEFA Bank GmbH and as a Research Assistant in the University of Wuppertal, AMNA group. After some experience from practice and academia, I was offered a PhD position in the same group, continuing in the same direction as my master thesis.
Why I am part of AMNA group as a PhD student
During my master thesis work, I was faced with the difficulties of pricing options via BSDE, especially in high dimensions. This triggered my attention to research more in this direction. Solution of high-dimensional BSDEs has been a long-lasting challenge in the community of Numerical and Applied Mathematics. However, with the advances of deep learning nowadays, this task is not far of being solved. Due to the connection of BSDEs to Partial Differential Equations, the solution of the stated problem gains much more attention and importance. Moreover, the application areas such as financial mathematics, optimal stochastic control, particle physics etc., makes it more important in practice. The opportunity to research in this direction was given to me from the AMNA department, which I accepted without hesitation.
What I will do during the PhD
The position I am working on is about solving high dimensional BSDEs using machine learning and GPU computing. Recent studies have shown that deep learning is the key to solving high dimensional BSDE and PDE problems. However, there is much more to do, especially in developing accurate and computationally efficient algorithms. In option pricing, high dimensional problems arise when pricing the value of a basket of stocks, where each stock represent one dimension. Considering real market properties, the structure of the BSDE that models such problems becomes more complex. One of the properties is to consider the risk of default. The methods developed so far fail when considering such cases. However, the deep learning field is truly diverse and offers a lot of opportunities to solve such problems efficiently, and the use of GPU computing reduces the computation cost. This is the goal of my PhD.