Lévy processes and quasi-shuffle algebras

Abstract

Accepted for publication in Stochastics: An International Journal of Probability and Stochastic ProcessesWe investigate the algebra of repeated integrals of semimartingales. We prove that a minimal family of semimartingales generates a quasi-shuffle algebra. In essence, to fulfill the minimality criterion, first, the family must be a minimal generator of the algebra of repeated integrals generated by its elements and by quadratic covariation processes recursively constructed from the elements of the family. Second, recursively constructed quadratic covariation processes may lie in the linear span of previously constructed ones and of the family, but may not lie in the linear span of repeated integrals of these. We prove that a finite family of independent Lévy processes that have finite moments generates a minimal family. Key to the proof are the Teugels martingales and a strong orthogonalization of them. We conclude that a finite family of independent Lévy processes form a quasi-shuffle algebra. We discuss important potential applications to constructing efficient numerical methods for the strong approximation of stochastic differential equations driven by Lévy processes.

Quasi‐shuffle algebras and renormalisation of rough differential equations

“…Taking the left endpoint results in an Itô integral, whilst taking the midpoint gives a Stratonovich integral. Different rough path lifts are related by a renormalisation procedure , for example, the usual Itô–Stratonovich conversion is given by The importance of deciding upon and relating such lifts in practical stochastic modelling has resulted in several alternative descriptions of this apparent ambiguity in defining the stochastic integral, namely (1)the canonical extensions of McShane and Marcus ; (2)Hoffman's exponential and logarithm ; (3)translations of rough paths by Bruned et al . . …”

Algebraic structures and stochastic differential equations driven by Lévy processes

“…I kuu+kv v = k u I u + k v I v , for any constants k u , k v ∈ R and words u, v ∈ A * . This was proved in Curry et al [21], it had already been established by Gaines [30] for drift-diffusions and Li & Liu [51] for jump-diffusions.…”

“…Remark 15 (Gradings preserved by quasi-shuffles) The power bracket grading introduced in Curry et al [21,Section 4], defined on uncompensated brackets, is grading preserving for any quasi-shuffle. This assigns the value 2 to the letter 0, the value 1 to letters 1, .…”

“…. , d. From Protter [73, p. 58] and Curry et al [21,Section 4] for words u, v and letters a, b from the extended alphabet A we have the following product rule…”

An isomorphism between branched and geometric rough paths

“…Remark 6.2. A series of papers [BCE18, CEFMW14,EFMPW15] have studied the relation between Itô and Stratonovich iterated integrals, particularity in relation with the quasi-shuffle algebra and Hoffman's exponential. One of the main results of [BCE18] is the existence of a unique a Hopf algebra morphism Ψ * : H * → H * whose adjoint is the arborification of the Hoffman exponential, see [BCE18, Thm.…”