Limit theorems for individual-based models in economics and finance. (English) Zbl 1168.60369
Summary: There is a widespread recent interest in using ideas from statistical physics to model certain types of problems in economics and finance. The main idea is to derive the macroscopic behavior of the market from the random local interactions between agents. Our purpose is to present a general framework that encompasses a broad range of models, by proving a law of large numbers and a central limit theorem for certain interacting particle systems with very general state spaces. To do this we draw inspiration from some work done in mathematical ecology and mathematical physics. The first result is proved for the system seen as a measure-valued process, while to prove the second one we will need to introduce a chain of embeddings of some abstract Banach and Hilbert spaces of test functions and prove that the fluctuations converge to the solution of a certain generalized Gaussian stochastic differential equation taking values in the dual of one of these spaces.
MSC:
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 60B12 | Limit theorems for vector-valued random variables (infinite-dimensional case) |
| 46N30 | Applications of functional analysis in probability theory and statistics |
| 62P20 | Applications of statistics to economics |
| 62P05 | Applications of statistics to actuarial sciences and financial mathematics |
| 91B70 | Stochastic models in economics |
Keywords:
individual-based model; interacting particle system; law of large numbers; central limit theorem; fluctuation process; measure-valued process; finance; economicsReferences:
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