Michael Grabchak

Sébastien Bossu, UNC Charlotte

Title: Integral regularization & theory of distributions: A gentle introduction

Andrew Papanicolaou, Carolina State University

Title: Principal Eigenportfolios and Primary Factors

Abstract: Multiple financial assets’ time-series data is stored in a matrix upon which we perform principal component analysis to find predominant factors in the market. Random matrix theory helps us to identify the number of factors present in the data, with the top eigenvalue-eigenvector pair bearing a strong resemblance to the market’s capitalization-weighted portfolio. This resemblance is consistent with fundamental concepts from portfolio theory, and can be extended to tensors of implied volatilities for which factors can be constructed using open interest as the analogue for capitalization. In our analyses we initially rely on the support of the Marchenko-Pastur distribution to serve as a cutoff for identification of outlying eigenvalues, but improved criteria can be developed using free probability.

Madhumita Paul, UNC Charlotte

Title: Erdos-Renyi model on spider-like quantum graphs

Xingnan Zhang, UNC Charlotte

Title: The Multivariate Dickman Distribution and Its Applications

Madhumita Paul, UNC Charlotte

Title: Brownian motion on spider type quantum graphs

Vlad Mărgărint, UNC Charlotte

Title: The Riemann Zeta Function, Cramer’s Model, and Dirichlet series

Stanislav Molchanov, UNC Charlotte

Title: Products of random matrices, Furstenberg Theorem, and the Dynamo problem

Stanislav Molchanov, UNC Charlotte

Title: Brownian motion on the periodic quantum graphs and related spectral problems

Jiaming Chen, NYU Shanghai

Title: New stochastic Fubini theorem of measure-valued processes via stochastic integration

Abstract:

Classical stochastic Fubini theorems start from a fixed semimartingale S, say, and a family of integrands for S which are parametrized by a parameter z, say, from some parameter space Z. There is a (non- random) measure μ, say, on Z, and the stochastic Fubini theorem then says that integration with respect to μ and stochastic integration with respect to S can be interchanged. In other words, we can either 1) first integrate the parametrized integrands with respect to μ and then stochastically integrate the resulting process with respect to S, or we can 2) first stochastically integrate, for each z, the corresponding integrand with respect to S and then integrate the result with respect to μ – and both double integrals yield the same result. What happens now if we replace the fixed measure μ by a stochastic kernel from the predictable sigma- field to Z? Approach 1) still makes sense, but how about 2)? And can we still get some kind of stochastic Fubini theorem? We show that we can, but we need to define for that a stochastic integral, with respect to S, of suitable measure-valued processes. The origin of this question comes from a (still open) question in mathematical finance. There are also some connections to a class of Volterra-type semimartingales. Based on joint work with Tahir Choulli and Martin Schweizer.

Stanislav Molchanov, UNC Charlotte

Title: Hierarchical Random Processes