Asymptotics for a Symmetric Equation in Price Formation

• Published in: Applied mathematics & optimization Vol. 59; no. 2; pp. 233 - 246
• Main Authors: González, María del Mar, Gualdani, Maria Pia
• Format: Journal Article
• Language: English
• Published: New York Springer-Verlag 01.04.2009; Springer Nature B.V
• Subjects: Asymptotic properties; ASYMPTOTIC SOLUTIONS; Calculus of Variations and Optimal Control; Optimization; Control; Decay; Derivatives; Diffusion; EQUATIONS; Evolution; Free boundaries; GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; Mathematical analysis; Mathematical and Computational Physics; MATHEMATICAL EVOLUTION; MATHEMATICAL METHODS AND COMPUTING; Mathematical Methods in Physics; Mathematical models; Mathematics; Mathematics and Statistics; MEAN-FIELD THEORY; Numerical and Computational Physics; ONE-DIMENSIONAL CALCULATIONS; Optimization; PRICES; Simulation; Studies; Systems Theory; Theoretical; Asymptotic decay; Dynamical equilibrium in price formation; Diffusion equation; Mean field model
• ISSN: 0095-4616; 1432-0606
• DOI: 10.1007/s00245-008-9052-y

We study the existence and asymptotics for large time of the solutions to a one dimensional evolution equation with non-standard right-hand side. The right-hand side involves the derivative of the solution computed at a given point. Existence is proven through a fixed point argument. When the problem is considered in a bounded interval, it is shown that the solution decays exponentially to the stationary state. This problem is a particular case of a mean-field free boundary model proposed by Lasry and Lions on price formation and dynamic equilibria.