Michel De Lara

Stochastic viability and dynamic programming

This paper deals with the stochastic control of nonlinear systems in the presence of state and control constraints, for uncertain discrete-time dynamics in finite dimensional spaces. In the deterministic case, the viability kernel is known to play a basic role for the analysis of such problems and the design of viable control feedbacks. In the present paper, we show how a stochastic viability kernel and viable feedbacks relying on probability (or chance) constraints can be defined and computed by a dynamic programming equation. An example illustrates most of the assertions.

Risk Aversion, Road Choice, and the One-Armed Bandit Problem

This paper provides a theoretical analysis of advanced traveler information systems for road choice with risk-averse drivers who rationally learn over time, in a simple setting. For this purpose, we study the one-armed bandit problem where a driver selects, day after day, either a safe or a random road. Four information regimes are envisaged. The visionary driver knows beforehand, with certainty, the travel time on the random road, while the locally informed driver needs to select a road to acquire information on it. Two intermediary information regimes (fully and globally) are also envisaged. We analyze these four regimes and compare the optimal strategies and the individual benefits with respect to individual risk aversion. A numerical example also illustrates the impact of risk aversion on dynamic optimal strategies.

Monotonicity properties for the viable control of discrete-time systems

This paper deals with the control of nonlinear systems in the presence of state and control constraints for discrete-time dynamics in finite-dimensional spaces. The viability kernel is known to play a basic role for the analysis of such problems and the design of viable control feedbacks. Unfortunately, this kernel may display very nonregular geometry and its computation is not an easy task in general. In the present paper, we show how monotonicity properties of both dynamics and constraints allow for relevant analytical upper and lower approximations of the viability kernel through weakly and strongly invariant sets. An example on fish harvesting management illustrates some of the assertions.

Ecosystem Viable Yields

The World Summit on Sustainable Development (Johannesburg, 2002) encouraged the application of the ecosystem approach by 2010. However, at the same summit, the signatory States undertook to restore and exploit their stocks at maximum sustainable yield (MSY), a concept and practice without ecosystemic dimension, since MSY is computed species by species, on the basis of a monospecific model. Acknowledging this gap, we propose a definition of “ecosystem viable yields” (EVY) as yields compatible (a) with biological safety levels (over which biomasses can be maintained for all times) and (b) with an ecosystem dynamics. The difference from MSY is that this notion is not based on equilibrium but on viability theory, which offers advantages for robustness. For a generic class of multispecies models with harvesting, we provide explicit expressions for the EVY. We apply our approach to the anchovy–hake couple in the Peruvian upwelling ecosystem.

Satisficing Versus Optimality: Criteria for Sustainability

Economic analysis addresses risk and long-term issues with discounted expected utility, focusing on optimality. Viability theory is rooted on satisfying sustainability constraints over time, focusing on feasibility. We build a bridge between these two approaches by establishing that viability is equivalent to an array of degenerate intertemporal optimization problems. First, we focus our attention on the deterministic case. We highlight the connections between the viability kernel and the minimum time of crisis. Carrying on, we lay out stochastic viability, turning the spotlight onto the notions of viable scenario and maximal viability probability. Our conceptual results bring the viability approach closer to the economic approach, especially in the stochastic case and regarding efficiency. We discuss the possible use of viability as a theoretical framework for biodiversity conservation, ecosystem management and climate change issues.

Reduction of the Zakai equation by invariance group techniques

A general procedure, inspired from that used for deterministic partial differential equations, is presented to reduce the Zakai stochastic Pde of filtering on n to a stochastic Pde on a lower-dimensional space m, with m

Building up time-consistency for risk measures and dynamic optimization

In stochastic optimal control, one deals with sequential decision-making under uncertainty; with dynamic risk measures, one assesses stochastic processes (costs) as time goes on and information accumulates. Under the same vocable of time-consistency (or dynamic-consistency), both theories coin two different notions: the latter is consistency between successive evaluations of a stochastic processes by a dynamic risk measure (a form of monotonicity); the former is consistency between solutions to intertemporal stochastic optimization problems. Interestingly, both notions meet in their use of dynamic programming, or nested, equations.

To what extent can ecosystem services motivate protecting biodiversity

Society increasingly focuses on managing nature for the services it provides people rather than for the existence of particular species. How much biodiversity protection would result from this modified focus? Although biodiversity contributes to ecosystem services, the details of which species are critical, and whether they will go functionally extinct in the future, are fraught with uncertainty. Explicitly considering this uncertainty, we develop an analytical framework to determine how much biodiversity protection would arise solely from optimising net value from an ecosystem service. Using stochastic dynamic programming, we find that protecting a threshold number of species is optimal, and uncertainty surrounding how biodiversity produces services makes it optimal to protect more species than are presumed critical. We define conditions under which the economically optimal protection strategy is to protect all species, no species, and cases in between. We show how the optimal number of species to protect depends upon different relationships between species and services, including considering multiple services. Our analysis provides simple criteria to evaluate when managing for particular ecosystem services could warrant protecting all species, given uncertainty. Evaluating this criterion with empirical estimates from different ecosystems suggests that optimising some services will be more likely to protect most species than others.

Robust Viable Analysis of a Harvested Ecosystem Model

The World Summit on Sustainable Development (Johannesburg, 2002) encouraged the adoption of an ecosystem approach. In this perspective, we propose a theoretical management framework that deals jointly with three issues: (i) ecosystem dynamics, (ii) conflicting issues of production and preservation, and (iii) robustness with respect to dynamics uncertainties. We consider a discrete-time two-species dynamic model, where states are biomasses and where two controls act as harvesting efforts of each species. Uncertainties take the form of disturbances affecting each species growth factors and are assumed to take their values in a known given set. We define the robust viability kernel as the set of initial species biomasses such that at least one harvesting strategy guarantees minimal production and preservation levels for all times, whatever the uncertainties. We apply our approach to the anchovy-hake couple in the Peruvian upwelling ecosystem. We find that accounting for uncertainty sensibly shrinks the deterministic viability kernel (without uncertainties). We comment on the management implications of comparing robust viability kernels (with uncertainties) and the deterministic one (without uncertainties).

A tight sufficient condition for Radner-Stiglitz nonconcavity in the value of information

This paper deals with the existence of a nonconcavity in the value of information, as was first explained by Radner and Stiglitz [A nonconcavity in the value of information, in: M. Boyer, R.E. Kihlstrom (Eds.), Bayesian Models in Economic Theory, Elsevier Science Publishers, Amsterdam, 1984, pp. 33-52 (Chapter 3)]. After defining infinitesimal information distance variationIIDV, we find that IIDV = 0 is sufficient for a zero marginal value of information at the null. This is a condition only on the information structure and in particular is independent of the decision makers preferences. This condition is tight: when IIDV > 0, there exists a payoff function for which the marginal value of information at the null is positive under general assumptions.