The examples of Martha Stewart, the recently convicted popular owner of a lifestyle company known for her house-and-garden TV shows, or of analysts of investment banks such as Citigroup chairman Weill recently sentenced to high fines by the US Security and Exchange Commission show that agents on financial markets use insider information to create utility advantages. In the first case additional information enables the insider to follow more clever investment strategies than uninformed traders. In the second case agents interested in concealing real information create an effective asymmetry. Both examples suggest that risks generated by the action of better informed traders for example among analysts or members of the executive board of companies bring losses to uninformed shareholders. For this reason in the US the SEC supervises and regulates insider trading since 1934. Critics of these regulations argue that the use of insider information by its executives can be pro¯table for a company as well as for the community of its shareholders.
In this project we deal with the mathematical modelling of financial markets with heterogeneous information. Relying upon concepts from a huge literature in theoretical economics, we study questions interesting both for the mathematics and economics point of view from the perspective of modern stochastic analysis, stochastic dynamics and the stochastic calculus of variations. The modern arsenal of these mathematical disciplines allows the modelling of insider trading beyond the framework given by Gaussian dynamics, for example by the stochastic analysis of Levy processes that permit price jumps. The toolbox of stochastic dynamics allows for a rigorous treatment of non-linear phenomena created for example in the context of herd behavior or coordinated action by informed traders.
In the first stage of our project we will continue the study of the close connection between utility and mathematical information theory in the framework of simple 2-agent models. For example we aim at taking into account dynamical utility functions as well as concepts of robust utility maximization, the counterparts of which on the information theoretic side are the minimal Shannon entropy or f-divergence. By means of data analysis and statistical methods of reduction of complexity we intend to identify insiders by their action on the market through their information drift.
In a second stage we will consider 3-agent models, the most common variant of which is due to Kyle and Back. In this context we plan to study problems with agents subject to conflicts of interest such as analysts, or experts such as climatologists with additional knowledge about catastrophes impacting the market.
On the long run we plan to treat non-linear effects in micro-economical equilibrium models of financial markets with meta-stable equilibrium states. For example we intend to interpret irreversible price shocks in the framework of hystereses in the stability diagram of a non-linear market.