Probability Seminar

Ion Grama, University of South Brittany

Title: Conditioned limit theorems for products of random matrices and Markov chains with applications to branching processes.

Abstract: Consider a random walk defined by the consecutive action of independent identically distributed random matrices on a starting point outside the unit ball in the d dimensional Euclidean space. We study the first moment when the walk enters the unit ball. We study the exact behavior of this time and prove conditioned limit theorems for the associated Markov walk. This extends to the case of walks on group GL(d,R) the well known results by Spitzer. The existence of the harmonic function related to the Markov walk turns out to be a crucial point of the proof. We have extended these results to general Markov chains and applied them to study the branching processes in Markov environment.

Isaac Sonin, UNC Charlotte

Title: A Continuous-Time Model of Financial Clearing Part 2

Abstract: We present a simple model of clearing in financial networks in continuous time. In the model, banks (firms, agents) are represented as tanks (reservoirs) with liquid (money) flowing in and out. This approach provides a simple recursive solution to a classical static model of financial clearing introduced by Eisenberg and Noe (2001). It also suggests a practical mechanism of simultaneous and real time payments. The dynamic structure of our model helps answer other related questions and, potentially, opens the way to handle more complicated dynamic financial networks, e.g., liabilities with different maturities. Also, our approach provides a useful tool for solving nonlinear equations involving a linear system and max min operations similar to the Bellman equation for the optimal stopping of Markov chains and other optimization problems.

Isaac Sonin, UNC Charlotte

Title: A Continuous-Time Model of Financial Clearing Part 1

Abstract: We present a simple model of clearing in financial networks in continuous time. In the model, banks (firms, agents) are represented as tanks (reservoirs) with liquid (money) flowing in and out. This approach provides a simple recursive solution to a classical static model of financial clearing introduced by Eisenberg and Noe (2001). It also suggests a practical mechanism of simultaneous and real time payments. The dynamic structure of our model helps answer other related questions and, potentially, opens the way to handle more complicated dynamic financial networks, e.g., liabilities with different maturities. Also, our approach provides a useful tool for solving nonlinear equations involving a linear system and max min operations similar to the Bellman equation for the optimal stopping of Markov chains and other optimization problems.

Stanislav Molchanov, UNC Charlotte

Title: Discrete Dynamo Part 4

Stanislav Molchanov, UNC Charlotte

Title: Discrete Dynamo Part 3

Stanislav Molchanov, UNC Charlotte

Title: Discrete Dynamo Part 2

Stanislav Molchanov, UNC Charlotte

Title: Discrete Dynamo