Methods for Solving and Analyzing Dynamic Models in the Face of Uncertainty and Cross-Sectional Heterogeneity

Competition in portfolio choice is a well documented phenomenon. In this talk, I will present optimal investment problems in Ito-diffusion markets for general forward utilities. I will discuss the notions of forward best-response and forward Nash equilibria for both cases of N-player games and at their continuum limit.

In this paper we investigate a path dependent optimal control problem on the process space with both drift and volatility controls, with possibly degenerate volatility. The dynamic value function is characterized by a fully nonlinear second order path dependent HJB equation on the process space, which is by nature infinite dimensional. In particular, our model covers mean field control problems with common noise as a special case. We shall introduce a new notion of viscosity solutions and establish both existence and comparison principle, under merely Lipschitz continuity assumptions. The main feature of our notion is that, besides the standard smooth part, the test function consists of an extra singular component which allows us to handle the second order derivatives of the smooth test functions without invoking the Ishii’s lemma. We shall use the doubling variable arguments, combined with the Ekeland-Borwein-Preiss Variational Principle in order to overcome the noncompactness of the state space. A smooth gauge-type function on the path space is crucial for our estimates.

“ Most of the literature studying heterogeneous-agent New Keynesian models takes household portfolios as exogenously given. What changes when agents are allowed to hedge aggregate risk? We develop a simple sequence-space method to solve for endogenous portfolios, impulse responses, and second-order risk premia in heterogeneous-agent models. Applying our method to a canonical HANK model, we show that the effects of monetary shocks and balanced-budget fiscal shocks are unchanged, but that unrestricted portfolio choice can greatly attenuate the response to deficit-financed fiscal shocks. The implied hedging portfolios, however, appear counterfactual, and imposing more realistic short-sale and leverage constraints restores standard outcomes.”

The main focus of this talk is the analysis of high-dimensional stochastic control problems in which many agents cooperate to minimize a convex cost functional. Our main results are sharp yet general bounds on the optimality gap between the full-information problem, in which each agent observes the states of all other agents, versus the distributed problem, in which each agent observes only its own state. Being decidedly non-asymptotic, our approach avoids structural constraints like exchangeability which are normally required in order to identify limiting objects, but which rule out network-based models. A protagonist in our approach, dubbed the “independent projection,” is the optimal approximation (in a precise sense) of a given high-dimensional diffusion process by one in which the coordinates are independent.