The 2015 AMMCS-CAIMS Congress

Interdisciplinary AMMCS Conference Series

Waterloo, Ontario, Canada | June 7-12, 2015

AMMCS-CAIMS 2015 Plenary Talk

Dependence between components of multivariate conditional Markov chains: Markov consistency and Markov Copulae

Tomasz Bielecki (Illinois Institute of Technology)

Modeling of evolution of dependence between processes occurring in financial markets is important. Typically, one can identify marginal statistical properties of individual processes, and then one is confronted with the task of modeling dependence between these individual processes so that the marginal properties are obeyed. We have been advocating, for some time now, to address this modeling problem via the theory of Markov consistency and Markov copulae.
In this talk we shall examine the problem of existence and construction of a non-trivial multivariate conditional Markov chain with components that are given conditional Markov chains. In this regard we shall give sufficient and necessary conditions, in terms of relevant conditional expectations, for a component of a multivariate Markov chain to be a Markov chain in the filtration of the entire chain — a property called strong Markov consistency, as well as in its own filtration — a property called weak Markov consistency. These characterization results are proved via analysis of the semi-martingale structure of the chain.
Several financial applications will be indicated.
Tomasz Bielecki is a Professor of Applied Mathematics and the Director of the Master of Mathematical Finance program at Illinois Institute of Technology. He received his PhD degree from the Warsaw School of Economics. Prof. Bielecki’s fields of expertise include Stochastic Analysis, Mathematical Finance, and Credit Risk Modeling. He is an Associate Editor of six well-known journals in areas of Mathematics and Finance, including Mathematical Finance and International Journal of Theoretical and Applied Finance. Prof. Bielecki is a co-author of three books in the area of Credit Risk Modeling and Financial Mathematics including his most recent book “Counterparty Risk and Funding: A Tale of Two Puzzles” co-authored with Stéphane Crépey and Damiano Brigo.