Title: Viscosity Characterizations of Explosions and Arbitrage
Description: This thesis analyzes the viscosity characterizations of the explosion time distribution for diffusions and of the arbitrage function in an equity market model with uncertainty. In the first part, we show that the tail distribution U of the explosion time for a multidimensional diffusion -- and more generally, a suitable function 𝒰 of the Feynman-Kac type involving the explosion time -- is a viscosity solution of an associated parabolic partial differential equation (PDE), provided that the dispersion and drift coefficients of the diffusion are continuous. This generalizes a result of Karatzas and Ruf (2015), who characterize U as a classical solution of a Cauchy problem for the PDE in the one-dimensional case, under the stronger condition of local Hölder continuity on the coefficients. We also extend their result to 𝒰 in the one-dimensional case by establishing the joint continuity of 𝒰. Furthermore, we show that 𝒰 is dominated by any nonnegative classical supersolution of this Cauchy problem.

Finally, we consider another notion of weak solvability, that of the distributional (sub/super)solution, and show that 𝒰 is no greater than any nonnegative distributional supersolution of the relevant PDE. In the second part, a more elaborate mathematical finance setting is taken. We show that, in an equity market model with Knightian uncertainty regarding the relative risk and covariance structure of its assets, the arbitrage function -- defined as the reciprocal of the highest return on investment that can be achieved relative to the market using nonanticipative strategies, and under any admissible market model configuration -- is a viscosity solution of an associated Hamilton-Jacobi-Bellman (HJB) equation under appropriate boundedness, continuity and Markovian assumptions on the uncertainty structure. This result generalizes that of Fernholz and Karatzas (2011), who characterized this arbitrage function as a classical solution of a Cauchy problem for this HJB equation under much stronger conditions than those needed here.

Our approach and results also extend to the Markovian Market Weight model introduced in Fernholz and Karatzas (2010b).