Nonlinear Differential Equations (DE) can explode in finite time. Generalizing DE to Backward Stochastic Differential Equations (BSDE), one can ask the following question: can we construct solutions that can explode in finite time to BSDE with nonlinear driver terms, i.e., can we specify +∞ as a possible value of the terminal condition of the BSDE? This and related questions have received considerable attention in the last decade, starting with the article (Popier, A. (2006), Backward stochastic differential equations with singular terminal condition, Stochastic Process. Appl., 116, 2014-2056) and found applications in mathematical finance, in particular, in the optimal liquidation of a portfolio of assets. Our 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.

Collaborators

• Ali Devin Sezer, Institute of Applied Mathematics, METU
• Çağın Ararat, Department of Industrial Engineering, Bilkent University
• Alexandre Popier,  Université du Maine 

Funded by  TUBITAK 1001.