• 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, ...
    • Cumulants in Non-commutative:Probability via Hopf Algebras 

      Rodriguez, Adrian de Jesus Celestino (Doctoral theses at NTNU;2022:370, Doctoral thesis, 2022)
      The notion of cumulants plays a significant role in the combinatorial study of noncommutative probability theory. In this thesis, we study several problems associated with the notions of cumulants for free, Boolean and ...
    • Cumulants, free cumulants and half-shuffles 

      Ebrahimi-Fard, Kurusch; Patras, Frederic (Peer reviewed; Journal article, 2015)
      Free cumulants were introduced as the proper analogue of classical cumulants in the theory of free probability. There is a mix of similarities and differences, when one considers the two families of cumulants. Whereas the ...
    • 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 ...
    • Magnus Expansion in Gauge Theories 

      Fredheim, Håkon Richard (Master thesis, 2022)
      Noen uttrykk for løsningen på en ordinær differensiallikning utledes ved hjelp av en metode som innebærer Taylor-rekkeutvikling av vektorfeltet som definerer differensiallikningen. En generalisering av et kjent resultat ...
    • 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 ...
    • Multiplicative and semi-multiplicative functions on non-crossing partitions, and relations to cumulants 

      Ebrahimi-Fard, Kurusch; Celestino, Adrián; Witzman, Leon; Perales, Daniel; Nica, Alexandru (Peer reviewed; Journal article, 2023)
      We consider the group (Ǧ, ∗)of unitized multiplicative functions in the incidence algebra of non-crossing partitions, where “∗” denotes the convolution operation. We introduce a larger group (Ǧ, ∗) of unitized functions ...
    • 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.
    • Post-Lie Magnus Expansion and BCH-Recursion 

      Al-Kaabi, Mahdi J. Hasan; Ebrahimi-Fard, Kurusch; Manchon, Dominique (Journal article; Peer reviewed, 2022)
    • Relations between infinitesimal non-commutative cumulants 

      Ebrahimi-Fard, Kurusch; de Jesus Celestino Rodr, Adrian; Perales Anaya, Daniel (Journal article; Peer reviewed, 2021)
      Boolean, free and monotone cumulants as well as relations among them, have proven to be important in the study of non-commutative probability theory. Quite notably, Boolean cumulants were successfully used to study free ...
    • 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 ...
    • Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability 

      Ebrahimi-Fard, Kurusch; Patra, Frédéric; Zambotti, Lorenzo; Tapia, Nikolas (Peer reviewed; Journal article, 2023)
      We study a particular group law on formal power series in non-commuting variables induced by their interpretation as linear forms on a suitable graded connected word Hopf algebra. This group law is left-linear and is ...
    • 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 ...
    • Tropical time series, iterated-sums signatures and quasisymmetric functions 

      Ebrahimi-Fard, Kurusch; Tapia, Nikolas; Diehl, Joscha (Journal article; Peer reviewed, 2022)
    • Universal Zero Dynamics: The SISO Case 

      Ebrahimi-Fard, Kurusch; Gray, W. Steven; Schmeding, Alexander (Chapter, 2021)
      Given a single-input, single-output (SISO) system with a Chen-Fliess series representation whose generating series has a well defined relative degree, it is shown that there is a notion of universal zero dynamics that ...
    • 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.
    • Wick polynomials in non-commutative probability: A group-theoretical approach 

      Ebrahimi-Fard, Kurusch; Patras, Frédéric; Tapia, Nikolas; Zambotti, Lorenzo (Journal article; Peer reviewed, 2021)
    • Zeroing the Output of Nonlinear Systems Without Relative Degree 

      Ebrahimi-Fard, Kurusch; Gray, W. Steven; Schmeding, Alexander (Book, 2023)
      The goal of this paper is to establish some facts concerning the problem of zeroing the output of an input-output system that does not have relative degree. The approach taken is to work with systems that have Chen-Fliess ...