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Bulletin of the London Mathematical Society

The Ito Formula for Perturbed Brownian Martingales

G. F. Vincent-Smith

Published Nov 1, 2000• 3 citations
Publisher

Abstract

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.

DOI
10.1112/s0024609300007712

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The quantum Ito formula has so far been proved for regular (bounded) quantum semimartingales We give three different extensions to classes of essentially self-adjoint (unbounded) quantum semimartingales. The first extension is to quantum semimartingales whose integrands have chaos matrices controlled by certain scalar matrices The second extension is to perturbations of classical, possibly unbounded, quantum semimartingales by bounded quantum semimartingales The third extension is obtained by perturbing an essentially self-adjoint quantum semimartingale satisfying the quantum Duhamel formula by a regular quantum semimartingle.
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