Panagiotis E. Souganidis, University of Chicago
Pathwise solutions for fully nonlinear first- and second-order partial differential equations with multiplicative rough time dependence
Abstract: The lectures are an overview of the theory for pathwise weak solutions to two classes of scalar fully nonlinear first- and second-order degenerate parabolic stochastic partial differential equations with multiplicative rough, a special case being Brownian, time dependence. These are Hamilton-Jacobi and Hamilton-Jacobi-Isaacs and quasilinear divergence form PDE including multidimensional scalar conservation laws. If the time dependence is “regular”, the weak solutions are respectively the viscosity and entropy/kinetic solutions. The main results are the well-posedness and qualitative properties of the solutions. Some concrete applications will also be discussed. The notes are mostly based on an ongoing collaboration with P.-L. Lions. Some of the work on conservation laws are also in collaboration with P.-L. Lions, B. Perthame and B. Gess.
Thaleia Zariphopoulou, University of Texas at Austin
PDE, BSDE and SPDE methods in optimal asset allocation
Abstract: In this lecture, I will present background results on optimal asset allocation and discuss how the associated stochastic optimisation problems are solved using elements from nonlinear PDE, BSDE and SPDE. Among others, I will present the classical Merton problem in portfolio management, indifference valuation in incomplete markets, and forward performance measurement in a variety of applications in real options, optimal execution, benchmarking, and others.
Prof. Jin Ma, University of Southern California
Backward Stochastic Differential Equations — Old and New
Abstract: The theory of Backward Stochastic Differential Equations (BSDEs) and its more general form, Forward-backward Stochastic Differential Equations (FBSDEs), have been studied extensively for the past 25 years, and are now ubiquitous in many branches of applied mathematics, in particular stochastic control theory and mathematical finance. In this lecture series I will give a brief overview of the existing literature of BSDEs and FBSDEs, and their relationship with various subjects in stochastic analysis and applications. In particular, starting from BSDE’s signature role in the so-called nonlinear Feynman-Kac formula, I will explore the relationship between BSDEs/FBSDEs with nonlinear PDEs, SPDEs, BSPDEs, as well as some newer related theories of nonlinear expectations, 2BSDEs, and path-dependent PDEs. Finally, I will briefly discuss the recently discovered role that the multidimensional BSDEs will likely to play in the study of time-inconsistent optimisation problems.