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Ph.D., University of Southern Denmark
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The first research objective is to prove the existence of PDE characterizations for a general class of optimization problems posed over a set of finitely-additive probability measures, i.e., over a subset of L1's bi-dual. Several interesting problems in math finance necessitate the use of finitely-additive measures instead of the countably-additive counterparts. Finitely-additive measures introduce a highly non-trivial singularity into the optimization problem. Together with a world leading expert in viscosity solutions and stochastic optimal control theory Mete Soner (ETH, Zurich) and long-term collaborator Gordan Zitkovic (UT-Austin), see the attached letters, we seek to prove a PDE characterization of the value function when the optimizer is a finitely-additive probability measure. Besides the areas of math finance and stochastic control theory, the proposed research will also impact the theory of asset pricing in incomplete markets since our formulation is highly related to marginal-utility-based pricing.
The second research objective is to prove the existence of equilibria in specific incomplete continuous-time and state models. The existing equilibrium theory requires model completeness, which implies that the individual investors are assumed able to trade their future uncertain income streams for upfront cash. Models in which investors cannot perform such a trade are deemed incomplete. Equilibrium theory has gained significant interest in the math finance community. However, incomplete models are mathematically tremendously complex and for the last 25 years the problem of proving equilibrium existence in incomplete continuous models has remained almost untouched. Therefore, the proposed research will impact the field of math finance. By its interdisciplinary nature the proposed research is also expected to impact the field of theoretical economics.