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arXiv · 2004.06945

Quasi-shuffle algebras in non-commutative stochastic calculus

Kurusch Ebrahimi-Fard·Frédéric Patras
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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.

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Kurusch Ebrahimi-Fard, Frédéric Patras. 2020-04-15. Quasi-shuffle algebras in non-commutative stochastic calculus. https://arxiv.org/abs/2004.06945

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