## Terry Rockafellar

#### General economic equilibrium with financial markets and

retainability.

**Abstract:** A theory of general economic equilibrium with incomplete financial markets is developed with many new features, including currency-denominated prices which enable treatment of currency-based derivative instruments and collateralized contracts. Prices in such model swith standard market structure have previously been articulated only in “units of account” which have no link to an actual currency and are subject to indeterminancy in scaling. That shortcoming, which prevents ordinary price comparisons between different states, present and future, has stemmed from a focus on consumption as the sole source of economic value, but here retention of goods is allowed to inuence their utility as well. The “goods” are not just commodities and thus can encompass other elements essential to finance. The framework is that of an economy operating in a currency agents nd attractive to retain, in balance with other needs. The attractiveness comes from Keynesian considerations about uncertainty which until now have not been brought in. An altered view of time and states helps by loosening the grip of perfect foresight in future markets. Existence is established with a single currency denominating the units of account in all states, and price indeterminancy is thereby removed. All contracts issued in the financial markets can be interpreted then as “real contracts.” Endogenously generated transaction costs on sales of contracts keep the financial markets from getting out of hand and lead to bid- ask spreads, including a gap between interest rates for lending and borrowing money. To this end, equilibrium is given a variational formulation that brings fresh tools to the subject. A different way of proving existence in that setting, not merely in a generic sense and without normalizing to a price simplex or arbitrarily xing “price levels” in the future states, makes use of duality bounds for the budget constraints. In the currency framework of the model, the proof of equilibrium is able moreover to proceed under far weaker assumptions than usual on the agents’ preferences and endowments.

## Roger Wets

#### Solving deterministic and stochastic equilibrium problems via augmented Walrasian.

**Abstract:** We described a method to solve deterministic and stochastic Walras equilibrium models based on associating with the equilibrium problem a bivariate function whose maxinf-points turn out to be Walras-equilibrium points. The numerical procedure relies on an augmentation of this bivariate function. Convergence of the proposedprocedure is proved by relying on lopsided convergence. In the dynamic and stochastic versions, we are mostly concerned with models that equip the agents with a mechanism to retain goods (savings) from one time period to the next but also allows for the transformation of goods via production. Preliminary results to models that include financial markets will be sketched out.

The presentation is based on work with Julio Deride (UC-Davis) and Alejandro Jofré (U. de Chile).

## Roger Guesnerie

#### Expectational coordination revisited: the “eductive” viewpoint, an overview.

**Abstract:** The standard economic viewpoint on expectational coordination has been challenged by the recent post 2008 events. The talk first reviews different existing directions of assessment of the rational expectations hypothesis that have been made to-date. It then argues that such a critical assessment, along the lines of the so-called “eductive” learning approach, radically modifies our view of three key problems: the economic role of speculation, the informational efficiency of markets and, last but not least, the ability of agents with long horizon to anticipate the future. It finally part stresses the future challenges of the approach.

## Teemu Pennanen

#### Convex duality in continuous-time stochastic optimization.

**Abstract:** We develop a duality framework for convex optimization problems over spaces of adapted stochastic processes. This is done by combining the conjugate duality theory of Rockafellar with some stochastic analysis. Various duality relations in stochastic control and mathematical finance are obtained as special cases. Besides classical models of financial markets, the general framework allows for e.g. illiquidity effects and portfolio constraints. This is joint work with Ari-Pekka Perkkiö.

## Jean Charles Rochet

#### A Lucas Model with Financial Frictions

**Abstract:** We introduce financial frictions into the Lucas (1978) “apple tree” model. Farmers produce fruit by renting land from landlords but they cannot borrow from financial markets. All farmers are subject to the same output shocks. To avoid bankruptcy, they store fruit as savings and adjust the scale of activity to the level of their precautionary savings. The land that is not rented to farmers is cultivated by land owners, but they are less productive. We show that there is a unique Markov competitive equilibrium, in which the rental price of land fluctuates in relation with the level of farmers’ savings. A decline in savings, caused by the adverse realization of output shocks, may bring the economy into a “poverty trap”.

## Jaksa Cvitanic

#### Moral Hazard in Dynamic Risk Management.

**Abstract:** We consider a contracting problem in which a principal hires an agent to manage a risky project. When the agent chooses volatility components of the output process and the principal observes the output continuously, the principal can compute the quadratic variation of the output, but not the individual components. This leads to moral hazard with respect to the risk choices of the agent. Using a recent theory of singular changes of measures for Ito processes, we formulate a principal-agent problem in this context, and solve it in the case of CARA preferences. In that case, the optimal contract is linear in these factors: the contractible sources of risk, including the output, the quadratic variation of the output and the cross-variations between the output and the contractible risk sources.

Thus, like sample Sharpe ratios used in practice, path-dependent contracts naturally arise when there is moral hazard with respect to risk management. We also provide comparative statics via numerical examples, showing that the optimal contract is sensitive to the values of risk premia and the initial values of the risk exposures.

## Bernard Cornet

#### Submodular financial markets with frictions.

**Abstract:** This paper characterizes financial markets with bid/ask spreads that are submodular, i.e., their super-replication cost functions (or super-hedging prices) are submodular, and studies an important class of submodular markets. The submodular assumption on the cost function, or the supermodularity usually assumed on preferences and utility functions, is the formal expression of perfect complementarity, that dates back to Fisher, Pareto, and Edgeworth, according to Samuelson (1974). Markets are all assumed to be arbitrage-free and to have a frictionless bond.

Our characterization result is two-fold. First, a market is submodular if and only if its super-replication cost is a Choquet integral and if and only if its set of risk-neutral probabilities is representable as the core of a submodular non-additive probability that is uniquely defined, called risk-neutral capacity. Second, a market is representable by its risk neutral capacity if and only if it is equivalent to a market, only composed of bid/ask event securities, i.e., whose payoffs are characteristic functions of events.

Our second contribution is to study an important class of submodular bid/ask financial markets, only composed of bid/ask event securities. Our main result shows that such a market is submodular if the events defining the securities are pairwise disjoint, hence in particular, if the event securities are bid/ask Arrow securities. The super-replication cost can then be calculated as a Choquet integral with respect to the risk-neutral capacity, that is given by an explicit and tractable formula.

Finally we will study the relationship between properties of functions defined on the whole space or the positive cone of a Riesz space, namely the classical properties of submodularity and convexity. Choquet (1954) conjectured that submodularity implies convexity for the important class of positively homogeneous functionals defined on the positive orthant of Rn, a result that has only been proved recently by K ̈onig (2003) and that is not true for more general Riesz spaces. Our study will replace the positive homogeneity property by translation invariance in the light of the work by Marinacci- Montrucchio (2008) and its application to financial markets.

## Sylvain Sorin

#### Dynamic games: link between discrete and continuous time.

**Abstract:** We will describe several examples showing the links in terms of concepts and tools between discrete and continuous time dynamic games. In particular there are new connections between repeated games and differential games.

We will discuss approachability theory, games with incomplete information and games with vanishing stage duration.

## Pierre Carpentier + Jean-Philippe Chancelier + Michel De Lara

#### Spatial Decomposition/Coordination Methods for Stochastic Optimal Control Problems.

**Abstract:** Power systems are becoming more and more complex, so that optimizing energy systems becomes more and more difficult. As optimization is challenged by the complexity due to large size, dynamical aspects and uncertainties, we claim that decomposition approaches may prove particularly adapted. This is why we present, in a unified framework, the main approaches to decompose multi-stage stochastic optimization problems for numerical resolution. This framework covers both Stochastic Programming (and scenario-based resolution methods like Progressive Hedging) and Stochastic Optimal Control (and state-based resolution methods like Stochastic Dynamic Programming), the two most well-known approaches and methods in multi-stage stochastic optimization.

This done, we go in more detail and outline the Dual Approximate Dynamic Programming (DADP) approach. DADP is a spatial decomposition method that solves an approximation of the original stochastic optimization problem. The approximation consists in relaxing an almost sure coupling constraint into its conditional expectation with respect to a given stochastic process. We discuss practical questions related to the implementation of the method and we illustrate DADP on the problem of managing a chain of hydroelectric dams. We finally discuss theoretical questions raised by the method.

## Juan Escobar

#### Cooperation dynamics.

**Abstract:** Cooperative relationships have a rich dynamics: Agents sometimes enter into non-cooperative phases, and sometimes, these non-cooperative phases end, and cooperation is restarted. In this paper, we study cooperation dynamics in long-term relations, and ask: Under what conditions does cooperation begin? Why does it end? And, how can it be restarted?

We answer these questions in the context of repeated games of incomplete information. Agents have private and imperfectly persistent private types. Actions are perfectly monitored and communication is not feasible. We characterize optimal equilibria.