On A Class Of Rank-Based Continuous Semimartingales

2021/04/09 by David Itkin, Martin Larsson, Itkin, David +1
Economics, Econometrics and Finance · Mathematics · #60J46 #60J60 (Primary) #70F10 (Secondary) #FOS: Economics and business #FOS: Mathematics #Financial Risk and Volatility Modeling #Mathematical Dynamics and Fractals #Mathematical Finance (q-fin.MF) #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2104.04396

openalex publication_date 2021/04/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Using the theory of Dirichlet forms we construct a large class of continuous semimartingales on an open domain E ⊂ ℝ^d, which are governed by rank-based, in addition to name-based, characteristics. Using the results of Baur et al. [Potential Analysis, 38(4):1233-1258,2013] we obtain a strong Feller property for this class of diffusions. As a consequence we are able to establish the nonexistence of triple collisions and obtain a simplified formula for the dynamics of its rank process. We also establish conditions under which the process is ergodic. Our main motivation is Stochastic Portfolio Theory (SPT), where rank-based diffusions of this type are used to model financial markets. We show that three main classes of models studied in SPT -- Atlas models, generalized volatility-stabilized models and polynomial models -- are special cases of our framework.

Related

• Convergence rates for rank-based models with applications to portfolio theory
• Stratonovich representation of semimartingale rank processes
• Chaoticity of the stationary distribution of rank-based interacting diffusions
• Long time behaviour and mean-field limit of Atlas models
• SPDE limit of the global fluctuations in rank-based models