AMC-SS: Risk, Ambiguity and the Long Run

Trustees Of Boston University, Boston

Investigators

Abstract

This research supported by this award focuses on optimization problems which combine uncertainty with intertemporal objectives. These issues are central to Asset Pricing applications, such as dynamic portfolio choice and derivatives pricing with several assets and state variables, for preference structures which include expected utility, recursive utility, and multiple priors. In their mathematical formulation, these models lead to complex control problems, involving multidimensional diffusions and stochastic differential equations of forward and backward type. A satisfactory analysis of these models requires explicit solutions, either exact or asymptotic, which disentangle the impact of price dynamics from those of the different aspects of preferences. The reserach will (i) develop a theory for long-run asymptotics, combining martingale methods, duality theory, Markov processes, and stochastic control; (ii) investigate the expansion of exact solutions around their long-run asymptotics, exploiting ideas of spectral theory; (iii) explore the connections to turnpike theorems, risk-sensitive control, and large deviations; and (iv) develop solutions to models of interest. This problem-driven research will seek new insights for Asset Pricing applications, which in turn suggest new directions for advancing mathematical theories. The involvement of graduate students in this effort will enhance their cross-training in Mathematical Finance and Applied Probability, and will bridge the traditional language gaps with related fields. This research has a twofold external relevance: first, a thorough understanding of portfolio choice is fundamental to devise efficient pension schemes, and to evaluate their welfare properties. Second, the equilibrium implications of long run investments are key to understanding the link between macroeconomic variables and asset price dynamics, and the interplay between risk aversion, ambiguity aversion and intertemporal substitution.