Series of iterated quantum stochastic integrals

R. L. Hudson, Sylvia Pulmannová

Motivated by the search for solutions of the quantum Yang–Baxter equation, an algebraic theory of quantum stochastic product integrals is developed. The product integrators are formal power series in an indeterminate h whose coefficients are elements of the Lie algebra ℒ labelling the usual integrators of a many-dimensional quantum stochastic calculus. The product integrals are also formal power series in h, whose coefficients are finite iterated additive stochastic integrals which act on the exponential domain in the Fock space of the calculus and which represent elements of the universal enveloping algebra 𝒰 of ℒ. They obey a multiplication rule suggested by the quantum Itô product formula, and are characterized among all such formal power series by a grouplike property.

R. F. Streater

Andrzej Łuczak

R. L. Hudson, Sylvia Pulmannová

A theory is developed of product integrals of the form ∏a<s<b ∏c<t<d(1+g[h] (ds,dt)). Here [a,b[ and [c,d[ are disjoint finite subintervals of R+, and g[h] is a formal power series in the indeterminate h whose constant term is zero and whose coefficients are elements of L⊗L, where ℒ is the space of basic differentials of a multidimensional quantum stochastic calculus. The product integrals are themselves formal power series in h whose coefficients are finite sums of iterated stochastic integrals against the elements of ℒ. They are symmetrized in such a way that ∏a<s<b ∏c<t<d(1+g[h](ds,dt)) is the image, obtained by applying the representation J[a,b[⊗J[c,d[ to the coefficients, where J[a,b[ is the representation canonically associated with the interval [a,b[, of a formal power series ∏ ∏(1+dg[h]) whose coefficients lie in U⊗U, where 𝒰 is the universal enveloping algebra of the Lie algebra ℒ. It is shown that the naturally conjectured multiplication rule, analogous to the multiplication rule for simple product integrals, holds in the commutative case.

C Barnett, R. F. Streater, Ivan F. Wilde

R. L. Hudson

We show that iterated stochastic integrals can be described equivalently either by the conventional forward adapted, or by backward adapted quantum stochastic calculus. By using this equivalence, we establish two properties of triangular (causal) and rectangular double quantum stochastic product integrals, namely a necessary and sufficient condition for their unitarity, and the coboundary relation between the former and the latter.