Functional limit theorems for financial markets with long-range dependence
Date: Friday, 21 May 2021
Time: 4-5 pm
Speaker: Professor Yuliya Mishura (Kyiv National University, Ukraine)
Abstract:
We start with an additive stochastic sequence that is based on the sequence of independent identically distributed (iid) random variables and has the coefficients that allow for dependency on the past. Then we formulate the conditions of the weak convergence to some limit process in terms of coefficients and characteristic function of any basic random variable. We adapt the general conditions to the case where the limit process is Gaussian. Then we move onto the multiplicative scheme in order to get a positive limit process (with the probability 1) that can be used for modelling of some asset prices from financial markets. Hence, we assume that all multipliers in the prelimit multiplicative scheme are positive, and therefore impose additional restrictions on the coefficients. In this talk, we will only consider Bernoulli's basic random variables. The next goal is to apply these general results to the case, where the limit processes in the additive schemes are fractional Brownian motion (fBm) and Riemann-Liouville fBm. In the case of the fBm limit, we consider the prelimit processes that are constructed using Cholesky decomposition of the covariance the function of fBm, and in both cases, we show that such coefficients are also suitable for the multiplicative scheme. The proofs are based on an in-depth study of the properties of the Cholesky decomposition of the covariance matrix of fBm.
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