Quasi-shuffle algebras in non-commutative stochastic calculus
Probability
2020-04-16 v1
Abstract
This chapter is divided into two parts. The first is largely expository and builds on Karandikar's axiomatisation of It{\^o} calculus for matrix-valued semimartin-gales. Its aim is to unfold in detail the algebraic structures implied for iterated It{\^o} and Stratonovich integrals. These constructions generalise the classical rules of Chen calculus for deterministic scalar-valued iterated integrals. The second part develops the stochastic analog of what is commonly called chronological calculus in control theory. We obtain in particular a pre-Lie Magnus formula for the logarithm of the It{\^o} stochastic exponential of matrix-valued semimartingales.
Keywords
Cite
@article{arxiv.2004.06945,
title = {Quasi-shuffle algebras in non-commutative stochastic calculus},
author = {Kurusch Ebrahimi-Fard and Frédéric Patras},
journal= {arXiv preprint arXiv:2004.06945},
year = {2020}
}