Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userQuasi-shuffle algebras in non-commutative stochastic calculus
69
0
0.0
(
0
)
Authors
Kurusch Ebrahimi-Fard
Ask ChatGPT about the research
No Arabic abstract
This chapter is divided into two parts. The first is largely expository and builds on Karandikars axiomatisation of It{^o} calculus for matrix-valued semimartin-gales. Its aim is to unfold in detail the algebraic structures implied for iterated It{^o} 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{^o} stochastic exponential of matrix-valued semimartingales.
rate research
Read More
We 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 Levy 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 Levy 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 Levy processes.
We discuss a non--commutative integration calculus arising in the mathematical description of anomalies in fermion--Yang--Mills systems. We consider the differential complex of forms $u_0ccr{eps}{u_1}cdotsccr{eps}{u_n}$ with $eps$ a grading operator on a Hilbert space $cH$ and $u_i$ bounded operators on $cH$ which naturally contains the compactly supported de Rham forms on $R^d$ (i.e. $eps$ is the sign of the free Dirac operator on $R^d$ and $cH$ a $L^2$--space on $R^d$). We present an elementary proof that the integral of $d$--forms $int_{R^d}trac{X_0dd X_1cdots dd X_d}$ for $X_iinMap(R^d;gl_N)$, is equal, up to a constant, to the conditional Hilbert space trace of $Gamma X_0ccr{eps}{X_1}cdotsccr{eps}{X_d}$ where $Gamma=1$ for $d$ odd and $Gamma=gamma_{d+1}$ (`$gamma_5$--matrix) a spin matrix anticommuting with $eps$ for $d$ even. This result provides a natural generalization of integration of de Rham forms to the setting of Connes non--commutative geometry which involves the ordinary Hilbert space trace rather than the Dixmier trace.
The Connes-Kreimer Hopf algebra of rooted trees, its dual, and the Foissy Hopf algebra of of planar rooted trees are related to each other and to the well-known Hopf algebras of symmetric and quasi-symmetric functions via a pair of commutative diagrams. We show how this point of view can simplify computations in the Connes-Kreimer Hopf algebra and its dual, particularly for combinatorial Dyson-Schwinger equations.
This paper shows how gauge theoretic structures arise naturally in a non-commutative calculus. Aspects of gauge theory, Hamiltonian mechanics and quantum mechanics arise naturally in the mathematics of a non-commutative framework for calculus and differential geometry. We show how a covariant version of the Levi-Civita connection arises naturally in this commutator calculus. This connection satisfies the formula $$Gamma_{kij} + Gamma_{ikj} = abla_{j}g_{ik} = partial_{j} g_{ik} + [g_{ik}, A_j].$$ and so is exactly a generalization of the connection defined by Hermann Weyl in his original gauge theory. In the non-commutative world $cal N$ the metric indeed has a wider variability than the classical metric and its angular holonomy. Weyls idea was to work with such a wider variability of the metric. The present formalism provides a new context for Weyls original idea.
Using time-reversal, we introduce a stochastic integral for zero-energy additive functionals of symmetric Markov processes, extending earlier work of S. Nakao. Various properties of such stochastic integrals are discussed and an It^{o} formula for Dirichlet processes is obtained.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
