Center for Mathematical Economics - Tag [talk]
Abstract: We study the problem of optimally managing an inventory with unknown demand trend. Our formulation leads to a stochastic control problem under partial observation, in which a Brownian motion with non-observable drift can be singularly controlled in both an upward and downward direction. We first derive the equivalent separated problem under full information, with state-space components given by the Brownian motion and the filtering estimate of its unknown drift, and we then completely solve this latter problem. Our approach uses the transition amongst three different but equivalent problem formulations, links between two-dimensional bounded-variation stochastic control problems and games of optimal stopping, and probabilistic methods in combination with refined viscosity theory arguments. We show substantial regularity of (a transformed version of) the value function, we construct an optimal control rule, and we show that the free boundaries delineating (transformed) action and inaction regions are bounded globally Lipschitz continuous functions. This is based on a joint work with Salvatore Federico (University of Genova) and Neofytos Rodosthenous (Queen Mary University of London).
Abstract: Motivated by the centenary of the publication of Frank Knight’s book „Risk, Uncertainty, and Profit“, we review some of his ideas and discuss how they are reflected in current research in finance and economics. We discuss three recent contributions including insurance premia, viability and arbitrage in financial markets, and market breakdown due to uncertainty in prices.
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