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Register a new userThe Baker-Campbell-Hausdorff formula via mould calculus
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The well-known Baker-Campbell-Hausdorff theorem in Lie theory says that the logarithm of a noncommutative product e X e Y can be expressed in terms of iterated commutators of X and Y. This paper provides a gentle introduction t{o} Ecalles mould calculus and shows how it allows for a short proof of the above result, together with the classical Dynkin explicit formula [Dy47] for the logarithm, as well as another formula recently obtained by T. Kimura [Ki17] for the product of exponentials itself. We also analyse the relation between the two formulas and indicate their mould calculus generalization to a product of more exponentials.
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We present a compact Baker-Campbell-Hausdorff-Dynkin formula for the composition of Lorentz transformations $e^{sigma_i}$ in the spin representation (a.k.a. Lorentz rotors) in terms of their generators $sigma_i$: $$ ln(e^{sigma_1}e^{sigma_2}) = tanh^{-1}left(frac{ tanh sigma_1 + tanh sigma_2 + frac12[tanh sigma_1, tanh sigma_2] }{ 1 + frac12{tanh sigma_1, tanh sigma_2} }right) $$ This formula is general to geometric algebras (a.k.a. real Clifford algebras) of dimension $leq 4$, naturally generalising Rodrigues formula for rotations in $mathbb{R}^3$. In particular, it applies to Lorentz rotors within the framework of Hestenes spacetime algebra, and provides an efficient method for composing Lorentz generators. Computer implementations are possible with a complex $2times2$ matrix representation realised by the Pauli spin matrices. The formula is applied to the composition of relativistic $3$-velocities yielding simple expressions for the resulting boost and the concomitant Wigner angle.
The operad of moulds is realized in terms of an operational calculus of formal integrals (continuous formal power series). This leads to many simplifications and to the discovery of various suboperads. In particular, we prove a conjecture of the first author about the inverse image of non-crossing trees in the dendriform operad. Finally, we explain a connection with the formalism of noncommutative symmetric functions.
The ring operations and the metric on $C(X)$ are extended to the set $mathbb{H}_{nf}(X)$ of all nearly finite Hausdorff continuous interval valued functions and it is shown that $mathbb{H}_{nf}(X)$ is both rationally and topologically complete. Hence, the rings of quotients of $C(X)$ as well as their metric completions are represented as rings of Hausdorff continuous functions.
An explicit martingale representation for random variables described as a functional of a Levy process will be given. The Clark-Ocone theorem shows that integrands appeared in a martingale representation are given by conditional expectations of Malliavin derivatives. Our goal is to extend it to random variables which are not Malliavin differentiable. To this end, we make use of Itos formula, instead of Malliavin calculus. As an application to mathematical finance, we shall give an explicit representation of locally risk-minimizing strategy of digital options for exponential Levy models. Since the payoff of digital options is described by an indicator function, we also discuss the Malliavin differentiability of indicator functions with respect to Levy processes.
This note reviews the Peano-Baker series and its use to solve the general linear system of ODEs. The account is elementary and self-contained, and is meant as a pedagogic introduction to this approach, which is well known but usually treated as a folklore result or as a purely formal tool. Here, a simple convergence result is given, and two examples illustrate that the series can be used explicitly as well.
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