Quasi-shuffle algebras in non-commutative stochastic calculus

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.