• Algebraic structures and stochastic differential equations driven by Levy processes 

      Curry, Charles Henry Alexander; Ebrahimi-Fard, Kurusch; Malham, Simon J. A.; Wiese, Anke (Peer reviewed; Journal article, 2019)
      We construct an efficient integrator for stochastic differential systems driven by Lévy processes. An efficient integrator is a strong approximation that is more accurate than the corresponding stochastic Taylor approximation, ...
    • The Faá di Bruno Hopf algebra for multivariable feedback recursions in the center problem for higher order Abel equations 

      Ebrahimi-Fard, Kurusch; Gray, W. Steven (Peer reviewed; Journal article, 2019)
      Poincaré’s center problem asks for conditions under which a planar polynomial system of ordinary differential equations has a center. It is well understood that the Abel equation naturally describes the problem in a ...
    • Hopf-algebraic Deformations of Products and Wick Polynomials 

      Ebrahimi-Fard, Kurusch; Patras, Frederic; Tapia, Nikolas; Zambotti, Lorenzo (Journal article; Peer reviewed, 2018)
      We present an approach to cumulant–moment relations and Wick polynomials based on extensive use of convolution products of linear functionals on a coalgebra. This allows, in particular, to understand the construction of ...
    • Monotone, free, and boolean cumulants: a shuffle algebra approach 

      Ebrahimi-Fard, Kurusch; Patras, Frederic (Journal article; Peer reviewed, 2018)
      The theory of cumulants is revisited in the “Rota way”, that is, by following a combinatorial Hopf algebra approach. Monotone, free, and boolean cumulants are considered as infinitesimal characters over a particular ...
    • On non-commutative stochastic exponentials 

      Curry, Charles Henry Alexander; Ebrahimi-Fard, Kurusch; Patras, Frederic (Journal article; Peer reviewed, 2019)
      Using non-commutative shuffle algebra, we outline how the Magnus expansion allows to define explicit stochastic exponentials for matrix-valued continuous semimartingales and Stratonovich integrals.
    • Renormalisation group for multiple zeta values 

      Ebrahimi-Fard, Kurusch; Manchon, Dominique; Singer, Johannes; Zhao, Jianqang (Journal article; Peer reviewed, 2018)
      Calculating multiple zeta values at arguments of any sign in a way that is compatible with both the quasi-shuffle product as well as meromorphic continuation, is commonly referred to as the renormalisation problem for ...
    • Shuffle group laws: applications in free probability 

      Ebrahimi-Fard, Kurusch; Patras, Frederic (Peer reviewed; Journal article, 2019)
      Commutative shuffle products are known to be intimately related to universal formulas for products, exponentials and logarithms in group theory as well as in the theory of free Lie algebras, such as, for instance, the ...
    • What is a post-Lie algebra and why is useful in geometric integration 

      Curry, Charles Henry Alexander; Ebrahimi-Fard, Kurusch; Munthe-Kaas, Hans (Journal article; Peer reviewed, 2018)
      We explain the notion of a post-Lie algebra and outline its role in the theory of Lie group integrators.