Paper detail

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{ô} calculus for matrix-valued semimartin-gales. Its aim is to unfold in detail the algebraic structures implied for iterated It{ô} 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{ô} stochastic exponential of matrix-valued semimartingales.