Robin Hudson's home page

Research interests

My research has always been mainly in quantum probability. Within that field over the last twenty-five years I have concentrated on quantum stochastic calculus (MRC 81 S 25).

Motivated by the latter and in particular the multiplication formula for iterated quantum stochastic integrals, in recent years I have studied the so-called Ito shuffle Hopf algebra, which is a non commutative generalisation of the shuffle product Hopf algebra which lends itself to the construction of deformation quantisations of Lie bialgebras. The principle mechanism for constructing these, also motivated from quantum stochastic calculus, is the double product integral, which can be regarded as a purely algebraic construction using formal power series. This permits quantisation of so-called "quasitriangular" Lie bialgebras whose Lie brackets are commutators in an associative algebra. In the future I hope to quantise more general Lie bialgebras by studying infinitesimal generators of deformation coproducts called deformation differential maps, again motivated by quantum stochastic calculus.

An analytic theory of double product integrals as operators in Fock spaces is also being developed. The "causal" or triangular version of these, in which the regions of integration are increasing simplexes in the real plane is relarted to a quantum extension of Girsanov's theorem and hence to a quantum version of the Black-Scholes model in finance. They may also provide a general model for causal interactions in noisy environments in quantum physics.

Another current interest is a family of classical stochastic processes corresponding to invariants of U(N), which are constructed from a noncommutative Capelli determinant using multidimensional quantum stochastic calculus, of which the Poisson process corresponds to the trace.

Some recent publications

In preparation

• (110) The antipode in the Ito Hopf algebra. (with S Pulmannová)

• (109) Quantum Girsanov theorem and Black-Scholes models.

• (108) Causal and rectangular double products in quantum stochastic calculus.

Submitted

• (107) From algebraic to analytic double product integrals, submitted to Proceedings of Nottingham Quantum Probaility Workshop.

• (106) Capelli processes, submitted to Probability Theory and Related Fields

• (105) Deformation coproducts and differential maps (with S Pulmannová), submitted to Studia Mathematica

Published

• (104) An analytic double product integral, pp 241-250 in "Quantum Probability and Infinite Dimensional Analysis" , Proceedings Levicao 2005 ed L Accardi et al, World Scientific (2007).

• (103) A double dilation constructed from a double product of rotaions, Markov Processes and Related Fields 13, 169-190 (2007)

• (102) Stop times in Fock space quantum probability, Stochastics 79, 383-391 (2007).

• (101) The Z₂-graded sticky shuffle product Hopf algebra, pp 237-243 in "Quantum Probability", ed M Bozejko, Banach Centre Publications 73 (2006).

• (100) Double product integrals and Enriquez quantisation of Lie bialgebras II: the quantum Yang-Baxter equation, Letters in Mathematical Physics 72, 211-224 (2005) (with S Pulmannová).

• (99) Ito calculus and quantisation of Lie bialgebras, Ann. Inst. H Poincaré-PR 41, 375-390 (2005).