Module Free White Noise Flows

The main result of this paper is to extend to Hilbert module level the proof of the inclusion of (non-Hamiltonian) stochastic differential equations based on free noise into the class of Hamiltonian equations driven by free white noise. To achieve this goal, free white noise calculus is extended to a trivial Hilbert module. The white noise formulation of the Ito table is radically different from the usual Itô tables, both classical and quantum and, combined with the Accardi–Boukas approach to Ito algebra, allows to drastically simplify calculations. Infinitesimal generators of Hilbert module free flows are characterized in terms of stochastic derivations from an initial algebra into a white noise Itô algebra. We prove that any such derivation is the difference of a ⋆-homomorphism and a trivial embedding.

A Hida–Malliavin white noise calculus approach to optimal control

The classical maximum principle for optimal stochastic control states that if a control [Formula: see text] is optimal, then the corresponding Hamiltonian has a maximum at [Formula: see text]. The first proofs for this result assumed that the control did not enter the diffusion coefficient. Moreover, it was assumed that there were no jumps in the system. Subsequently, it was discovered by Shige Peng (still assuming no jumps) that one could also allow the diffusion coefficient to depend on the control, provided that the corresponding adjoint backward stochastic differential equation (BSDE) for the first-order derivative was extended to include an extra BSDE for the second-order derivatives. In this paper, we present an alternative approach based on Hida–Malliavin calculus and white noise theory. This enables us to handle the general case with jumps, allowing both the diffusion coefficient and the jump coefficient to depend on the control, and we do not need the extra BSDE with second-order derivatives. The result is illustrated by an example of a constrained linear-quadratic optimal control.

WHITE NOISE ANALYSIS: SOME APPLICATIONS IN COMPLEX SYSTEMS, BIOPHYSICS AND QUANTUM MECHANICS

The white noise calculus originated by T. Hida is presented as a powerful tool in investigating physical and social systems. Combined with Feynman's sum-over-all histories approach, we parameterize paths with memory of the past, and evaluate the corresponding probability density function. We discuss applications of this approach to problems in complex systems and biophysics. Examples in quantum mechanics with boundaries are also given where Markovian paths are considered.