In a further generalisation of the classical Ito formula, we show that if B = ( A t + A t † : t ∈ ] 0 , 1 [ ) is the Brownian quantum martingale and J = (Jt: t ∈ [0, 1]) is a bounded quantum semimartingale, then M = (Bt Jt: t ∈ [0, 1]) satisfies the functional quantum Ito formula [10, Section 6] [formula] The proof, which relies on the classical theory of unbounded self-adjoint operators, may be adapted to the case where B is replaced by a classical Brownian martingale. 2000 Mathematics Subject Classification 81S25, 60J65, 47D03.
We develop the theory of chaos spaces and chaos matrices. A chaos space is a Hilbert space with a fixed, countably-infinite, direct-sum decomposition. A chaos matrix between two chaos spaces is a doubly-infinite matrix of bounded operators which respects this decomposition. We study operators represented by such matrices, particularly with respect to self-adjointness. This theory is used to re-formulate the quantum stochastic calculus of Hudson and Parthasarathy. Integrals of chaos-matrix processes are defined using the Hitsuda-Skorokhod integral and Malliavin gradient,following Lindsay and Belavkin. A new way of defining adaptedness is developed and the consequent quantum product Ito formula is used to provide a genuine functional Ito formula for polynomials in a large class of unbounded processes, which include the Poisson process and Brownian motion. A new type of adaptedness, known as
Ω-adaptedness, is defined. We show that quantum stochastic integrals of
Ω-adapted processes are well-behaved; for instance, bounded processes have bounded integrals. We solve the appropriate modification of the evolution equation of Hudson and Parthasarathy: $U(t)=I+\int_{0}^{t}E(s)\mathrm{d}\Lambda(s)+F(s)\mathrm{d} A(s)+ G(s)U(s)\mathrm{d} A^{\dagger}(s)+H(s)U(s)\mathrm{d} s, $ where the coefficients are time-dependent, bounded,
Ω-adapted processes acting on the whole Fock space. We show that the usual conditions on the coefficients, viz. $(E,F,G,H)=(W-I,L,-WL^{*},iK+\mbox{$\frac{1}{2}$}LL^{*})$ where
W is unitary and
K self-adjoint, are necessary and sufficient conditions for the solution to be unitary. This is a very striking result when compared to the adapted case.
Using the Wiener–Poisson isomorphism, we show that if (Ft)0 ⩽ t ⩽ 1 is a real, bounded, predictable process adapted to the filtration of a compensated Poisson process (Xt)0 ⩽ t ⩽ 1, and if M ^ t is the operator corresponding to multiplication by M t = ∫ 0 t F s d X s , then for any regular self-adjoint quantum semimartingale J = ( J t ) 0 ⩽ t ⩽ 1 , the essentially self-adjoint quantum semimartingale ( M ^ t + J t ) 0 ⩽ t ⩽ 1 satisfies the quantum Ito formula. We also introduce a generalisation of the Poisson process to a measure space (M, M, μ) as an isometry I: L2 (M, M, μ) → L2(Ω, F, P) and give a new construction of the generalised Wiener–Poisson isomorphism WI: F+ (L2(M)) → L2 (Ω, F, P) using exponential vectors. Using C*-algebra theory, given any measure space we construct a canonical generalised Poisson process. Unlike other constructions, we make no a priori use of Poisson measures. 2000 Mathematics Subject Classification 60G20, 60G35, 46L53, 81S25.