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A non-local free boundary problem arising in a theory of financial bubbles.

ABSTRACT: We consider an evolution non-local free boundary problem that arises in the modelling of speculative bubbles. The solution of the model is the speculative component in the price of an asset. In the framework of viscosity solutions, we show the existence and uniqueness of the solution. We also show that the solution is convex in space, and establish several monotonicity properties of the solution and of the free boundary with respect to parameters of the problem. To study the free boundary, we use, in particular, the fact that the odd part of the solution solves a more standard obstacle problem. We show that the free boundary is [Formula: see text] and describe the asymptotics of the free boundary as c, the cost of transacting the asset, goes to zero.

SUBMITTER: Berestycki H

PROVIDER: S-EPMC4186254 | biostudies-literature | 2014 Nov

REPOSITORIES: biostudies-literature

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A non-local free boundary problem arising in a theory of financial bubbles.

Berestycki Henri H Monneau Regis R Scheinkman José A JA

Philosophical transactions. Series A, Mathematical, physical, and engineering sciences 20141101 2028

We consider an evolution non-local free boundary problem that arises in the modelling of speculative bubbles. The solution of the model is the speculative component in the price of an asset. In the framework of viscosity solutions, we show the existence and uniqueness of the solution. We also show that the solution is convex in space, and establish several monotonicity properties of the solution and of the free boundary with respect to parameters of the problem. To study the free boundary, we us ...[more]

PMID: 25288815

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Project description:Standard evolutionary theories of aging and mortality, implicitly based on assumptions of spatial averaging, hold that natural selection cannot favor shorter lifespan without direct compensating benefit to individual reproductive success. However, a number of empirical observations appear as exceptions to or are difficult to reconcile with this view, suggesting explicit lifespan control or programmed death mechanisms inconsistent with the classic understanding. Moreover, evolutionary models that take into account the spatial distributions of populations have been shown to exhibit a variety of self-limiting behaviors, maintained through environmental feedback. Here we extend recent work on spatial modeling of lifespan evolution, showing that both theory and phenomenology are consistent with programmed death. Spatial models show that self-limited lifespan robustly results in long-term benefit to a lineage; longer-lived variants may have a reproductive advantage for many generations, but shorter lifespan ultimately confers long-term reproductive advantage through environmental feedback acting on much longer time scales. Numerous model variations produce the same qualitative result, demonstrating insensitivity to detailed assumptions; the key conditions under which self-limited lifespan is favored are spatial extent and locally exhaustible resources. Factors including lower resource availability, higher consumption, and lower dispersal range are associated with evolution of shorter lifespan. A variety of empirical observations can parsimoniously be explained in terms of long-term selective advantage for intrinsic mortality. Classically anomalous empirical data on natural lifespans and intrinsic mortality, including observations of longer lifespan associated with increased predation, and evidence of programmed death in both unicellular and multicellular organisms, are consistent with specific model predictions. The generic nature of the spatial model conditions under which intrinsic mortality is favored suggests a firm theoretical basis for the idea that evolution can quite generally select for shorter lifespan directly.

Dataset Information

A non-local free boundary problem arising in a theory of financial bubbles.

Publications

A non-local free boundary problem arising in a theory of financial bubbles.

Similar Datasets

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