A class of orthogonal integrators for stochastic differential equations

The purpose of this paper is to construct a class of orthogonal integrators for stochastic differential equations (SDEs). The family of SDEs with orthogonal solutions is univocally characterized. For this, a class of orthogonal integrators is introduced by imposing constraints to Runge–Kutta (RK) matrices and weights of the standard stochastic RK schemes.The performance of the method is illustrated by means of numerical simulations.     

Nonlinear stochastic integration with a non-smooth family of integrators

We consider nonlinear stochastic integrals of Itô-type w.r.t. a family of semimartingales which depend on a spatial parameter. These integrals were introduced by Carmona/Nualart, Kunita, and Le Jan. The extension of the elementary nonlinear integral is based on the condition that the semimartingale kernel has nice continuity properties in the spatial parameter. We investigate the case that continuity is not available and suggest different directions of generalization. This brings us beyond the case that any integral can be approximated by integrals with integrands taking only finitely many values.     

Numerical integrators for quantum dynamics close to the adiabatic limit

Summary.  We study the numerical solution of singularly perturbed Schr?-dinger equations with time-dependent Hamiltonian. Based on a reformulation of the equations, we derive time-reversible numerical integrators which can be used with step sizes that are substantially larger than with traditional integration schemes. A complete error analysis is given for the adiabatic case. To deal with avoided crossings of energy levels, which lead to non-adiabatic behaviour, we propose an adaptive extension of the methods which resolves the sharp transients that appear in non-adiabatic state transitions. Received November 12, 2001 / Revised version received May 8, 2002 / Published online October 29, 2002 Mathematics Subject Classification (1991): 65L05, 65M15, 65M20, 65L70.     

On quantum stochastic differential equations

Existence and uniqueness theorems for quantum stochastic differential equations with nontrivial initial conditions are proved for coefficients with completely bounded columns. Applications are given for the case of finite-dimensional initial space or, more generally, for coefficients satisfying a finite localisability condition. Necessary and sufficient conditions are obtained for a conjugate pair of quantum stochastic cocycles on a finite-dimensional operator space to strongly satisfy such a quantum stochastic differential equation. This gives an alternative approach to quantum stochastic convolution cocycles on a coalgebra.     

Strong approximation of time-changed stochastic differential equations involving drifts with random and non-random integrators

BIT Numerical Mathematics - The rates of strong convergence for various approximation schemes are investigated for a class of stochastic differential equations (SDEs) which involve a random time...     

A new form and a ⋆-algebraic structure of quantum stochastic integrals in Fock space

An algebraic definition of the basic quantum process for the noncommutative stochastic calculus is given in terms of the Fock representation of a Lie ⋆-algebra of matrices in a pseudo-Euclidean space. An operator definition of the quantum stochastic integral is given and its continuity is proved in a projective limit uniform operator topology. A new form of quantum stochastic equations, revealing the ⋆-algebraic structure of quantum Ito's formula, is given. (Conferenza tenuta il 21 settembre 1988)     

Malliavin calculus for quantum stochastic processes

A Girsanov formula and an integration by parts formula are given for quantum stochastic processes on the Heisenberg-Weyl algebra and used to obtain sufficient conditions for their Wigner density in a given state to lie in the Sobolev space of order k.     

A quantum stochastic Lie-Trotter product formula

A Trotter product formula is established for unitary quantum stochastic processes governed by quantum stochastic differential equations with constant bounded coefficients.     

Quantum stochastic calculus and quantum Gaussian processes

In this lecture we present a brief outline of boson Fock space stochastic calculus based on the creation, conservation and annihilation operators of free field theory, as given in the 1984 paper of Hudson and Parthasarathy [9]. We show how a part of this architecture yields Gaussian fields stationary under a group action. Then we introduce the notion of semigroups of quasifree completely positive maps on the algebra of all bounded operators in the boson Fock space Γ(? ⁿ ) over ? ⁿ . These semigroups are not strongly continuous but their preduals map Gaussian states to Gaussian states. They were first introduced and their generators were shown to be of the Lindblad type by Vanheuverzwijn [19]. They were recently investigated in the context of quantum information theory by Heinosaari et al. [7]. Here we present the exact noisy Schrödinger equation which dilates such a semigroup to a quantum Gaussian Markov process.     

Sesquilinear quantum stochastic analysis in Banach space

A theory of quantum stochastic processes in Banach space is initiated. The processes considered here consist of Banach space valued sesquilinear maps. We establish an existence and uniqueness theorem for quantum stochastic differential equations in Banach modules, show that solutions in unital Banach algebras yield stochastic cocycles, give sufficient conditions for a stochastic cocycle to satisfy such an equation, and prove a stochastic Lie–Trotter product formula. The theory is used to extend, unify and refine standard quantum stochastic analysis through different choices of Banach space, of which there are three paradigm classes: spaces of bounded Hilbert space operators, operator mapping spaces and duals of operator space coalgebras. Our results provide the basis for a general theory of quantum stochastic processes in operator spaces, of which Lévy processes on compact quantum groups is a special case.     

The characterization of certain quantum stochastic stationary process

In this paper we will obtain a Stone type theorem under the frame of Hilbert C＊-module, such that the classical Stone theorem is our special case. Then we use it as a main tool to obtain a spectrum decomposition theorem of certain stationary quantum stochastic process. In the end, we will give it an interpretation in statistical mechanics of multi-linear response.     

Application of quantum stochastic calculus to optimal control

Given a plant whose output is described by a Hudson-Parthasarathy quantum stochastic differential equation [1–3] driven by standard quantum Brownian motion, we compute explicitly the control process that rapidly makes the size of the plant's output small, and keeps the energy used at a minimum. The solution to the quantum stochastic analogue of the linear regulator problem of classical stochastic control theory ([4], [5]) follows as a special case.Published in Matematicheskie Zametki, Vol. 53, No. 5, pp. 48–56, May, 1993.     

Noncommutative stochastic processes with independent and stationary increments satisfy quantum stochastic differential equations

Summary The notion of a unitary noncommutative stochastic process with independent and stationary increments is introduced, and it is proved that such a process, under a continuity assumption, can be embedded into the solution of a quantum stochastic differential equation in the sense of Hudson and Parthasarathy [8].This work was supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 123, Stochastische Mathematische Modelle