Quantum Black Noise and Generalized Quantum Stochastic Equations

Will Hicks

The Accardi-Boukas quantum Black-Scholes equation (Luigi Accardi, Andreas Boukas: The Quantum Black-Scholes Equation, Global Journal of Pure and Applied Mathematics, (2006) vol.2, no.2, pp. 155-170.) can be used as an alternative to the classical approach to finance, and has been found to have a number of useful benefits. The quantum Kolmogorov backward equations, and associated quantum Fokker-Planck equations, that arise from this general framework, are derived using the Hudson-Parthasarathy quantum stochastic calculus (Hudson, R.L.; Parthasarathy, K.R: Quantum Ito's Formula and Stochastic Evolutions. Commun Math. Phys. 1984, 93, 301--323.). In this paper we show how these equations can be derived using a nonlocal approach to quantum mechanics. We show how nonlocal diffusions, and quantum stochastic processes can be linked, and discuss how moment matching can be used for deriving solutions.

V. P. Belavkin, Jaeseong Heo, Un Cig Ji

Will Hicks

In this article we model a financial derivative price as an observable on the market state function. We apply geometric techniques to integrating the Heisenberg Equation of Motion. We illustrate how the non-commutative nature of the model introduces quantum interference effects that can act as either a drag or a boost on the resulting return. The ultimate objective is to investigate the nature of quantum drift in the Accardi-Boukas quantum Black-Scholes framework which involves modelling the financial market as a quantum observable, and introduces randomness through the Hudson-Parthasarathy quantum stochastic calculus. In particular we aim to differentiate randomness that is introduced through external noise (quantum stochastic calculus) and randomness that is fundamental to a quantum system (Heisenberg Equation of Motion).

Nobuaki Obata

Will Hicks

In this article we model a financial derivative price as an observable on a market state function, with a view to understanding how some of the non-commutative behaviour of the financial market impacts the dynamics.We integrate the Heisenberg Equation of Motion, by using a Riemannian metric, and illustrate how the non-commutative nature of the model introduces quantum interference effects that can act as either a drag or a boost on the resulting return.The ultimate objective is to investigate the nature of quantum drift in the Accardi-Boukas quantum Black-Scholes framework ([1]) which involves modelling the financial market as a quantum observable, and introduces randomness through the Hudson-Parthasarathy quantum stochastic calculus ([15]).In particular we aim to differentiate between randomness that is introduced through external noise (quantum stochastic calculus) and randomness that is intrinsic to a quantum system (Heisenberg Equation of Motion).

Crispin W. Gardiner