The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a self-contained exposition of the basic theory of viscosity solutions

https://www.ams.org/bull/1992-27-01/S0273-0979-1992-00266-5/S0273-0979-1992-00266-5.pdf

User’s guide to viscosity solutions of second order partial differential equations

As GDP is a measure of market capacity and not economic well-being, this report has been commissioned to more accurately understand the social progress indicators of any given state. Gross domestic product (GDP) is the most widely used measure of economic activity. There are international standards for its calculation, and much thought has gone into its statistical and conceptual bases. But GDP mainly measures market production, though it has often been treated as if it were a measure of economic well-being. Conflating the two can lead to misleading indications about how well-off people are and entail the wrong policy decisions. One reason why money measures of economic performance and living standards have come to play such an important role in our societies is that the monetary valuation of goods and services makes it easy to add up quantities of a very different nature. When we know the prices of apple juice and DVD players, we can add up their values and make statements about production and consumption in a single figure. But market prices are more than an accounting device. Economic theory tells us that when markets are functioning properly, the ratio of one market price to another is reflective of the relative appreciation of the two products by those who purchase them. Moreover, GDP captures all final goods in the economy, whether they are consumed by households, firms or government. Valuing them with their prices would thus seem to be a good way of capturing, in a single number, how well-off society is at a particular moment. Furthermore, keeping prices unchanged while observing how quantities of goods and services that enter GDP move over time would seem like a reasonable way of making a statement about how society’s living standards are evolving in real terms. As it turns out, things are more complicated. First, prices may not exist for some goods and services (if for instance government provides free health insurance or if households are engaged in child care), raising the question of how these services should be valued. Second, even where there are market prices, they may deviate from society’s underlying valuation. In particular, when the consumption or production of particular products affects society as a whole, the price that individuals pay for those products will differ from their value to society at large. Environmental damage caused by production or consumption activities that is not reflected in market prices is a well-known example.

https://andrewoswald.com/docs/stiglitz20report_commission20final20200920sept.pdf

Report by the commission on the measurement of economic performance and social progress

Notation.- Notation.- Gradient Flow in Metric Spaces.- Curves and Gradients in Metric Spaces.- Existence of Curves of Maximal Slope and their Variational Approximation.- Proofs of the Convergence Theorems.- Uniqueness, Generation of Contraction Semigroups, Error Estimates.- Gradient Flow in the Space of Probability Measures.- Preliminary Results on Measure Theory.- The Optimal Transportation Problem.- The Wasserstein Distance and its Behaviour along Geodesics.- Absolutely Continuous Curves in p(X) and the Continuity Equation.- Convex Functionals in p(X).- Metric Slope and Subdifferential Calculus in (X).- Gradient Flows and Curves of Maximal Slope in p(X).

Gradient Flows: In Metric Spaces and in the Space of Probability Measures

We survey here some recent studies concerning what we call mean-field models by analogy with Statistical Mechanics and Physics. More precisely, we present three examples of our mean-field approach to modelling in Economics and Finance (or other related subjects...). Roughly speaking, we are concerned with situations that involve a very large number of “rational players” with a limited information (or visibility) on the “game”. Each player chooses his optimal strategy in view of the global (or macroscopic) informations that are available to him and that result from the actions of all players. In the three examples we mention here, we derive a mean-field problem which consists in nonlinear differential equations. These equations are of a new type and our main goal here is to study them and establish their links with various fields of Analysis. We show in particular that these nonlinear problems are essentially well-posed problems i.e., have unique solutions. In addition, we give various limiting cases, examples and possible extensions. And we mention many open problems.

Mean Field Games

This article considers how experience can influence the developing and mature brain and proposes a new categorization scheme based upon the type of information stored and the brain mechanisms that appear to be involved in storing it. In this scheme, experience-expectant information storage refers to incorporation of environmental information that is ubiquitous in the environment and common to all species members, such as the basic elements of pattern perception. Experience-expectant processes appear to have evolved as a neural preparation for incorporating specific information: in many sensory systems, synaptic connections between nerve cells are overproduced, and a subsequent selection process occurs in which aspects of sensory experience determine the pattern of connections that remains. Experience-dependent information storage refers to incorporation of environmental information that is idiosyncratic, or unique to the individual, such as learning about one's specific physical environment or vocabulary. The neural basis of experience-dependent processes appears to involve active formation of new synaptic connections in response to the events providing the information to be stored. Although these processes probably do not occur entirely independently of one another in development, the categories offer a new view more in accord with neural mechanisms than were terms like "critical" or "sensitive period."

Experience and brain development.

Jeux à champ moyen. I – Le cas stationnaire

Jeux à champ moyen. II – Horizon fini et contrôle optimal

This paper presents an original probabilistic method for the numerical computations of Greeks (i.e. price sensitivities) in finance. Our approach is based on the {\it integration-by-parts} formula, which lies at the core of the theory of variational stochastic calculus, as developed in the Malliavin calculus. The Greeks formulae, both with respect to initial conditions and for smooth perturbations of the local volatility, are provided for general discontinuous path-dependent payoff functionals of multidimensional diffusion processes. We illustrate the results by applying the formula to exotic European options in the framework of the Black and Scholes model. Our method is compared to the Monte Carlo finite difference approach and turns out to be very efficient in the case of discontinuous payoff functionals.

http://janroman.dhis.org/finance/Malliavin%20Calculus/Malliavin%20and%20MC2.pdf

Applications of Malliavin calculus to Monte-Carlo methods in finance. II

This text is inspired from a "Cours Bachelier" held in January 2009 and taught by Jean-Michel Lasry. This course was based upon the articles of the three authors and upon unpublished materials developed by the authors. Proofs were not presented during the conferences and are now available. So are some issues that were only rapidly tackled during class. The content of this text is therefore far more important than the actual "Cours Bachelier" conferences, though the guiding principle is the same and consists in a progressive introduction of the concepts, methodologies and mathematical tools of mean field games theory.

Jean-Michel Lasry

Papers

Mean Field Games

Jeux à champ moyen. I – Le cas stationnaire

Jeux à champ moyen. II – Horizon fini et contrôle optimal

Applications of Malliavin calculus to Monte-Carlo methods in finance. II

Mean Field Games and Applications