1,904 research outputs found
Cram\'er-Rao bound for time-continuous measurements in linear Gaussian quantum systems
We describe a compact and reliable method to calculate the Fisher information
for the estimation of a dynamical parameter in a continuously measured linear
Gaussian quantum system. Unlike previous methods in the literature, which
involve the numerical integration of a stochastic master equation for the
corresponding density operator in a Hilbert space of infinite dimension, the
formulas here derived depends only on the evolution of first and second moments
of the quantum states, and thus can be easily evaluated without the need of any
approximation. We also present some basic but physically meaningful examples
where this result is exploited, calculating analytical and numerical bounds on
the estimation of the squeezing parameter for a quantum parametric amplifier,
and of a constant force acting on a mechanical oscillator in a standard
optomechanical scenario.Comment: 9 pages, 2 figure
Monte Carlo Euler approximations of HJM term structure financial models
We present Monte Carlo-Euler methods for a weak approximation problem related
to the Heath-Jarrow-Morton (HJM) term structure model, based on \Ito stochastic
differential equations in infinite dimensional spaces, and prove strong and
weak error convergence estimates. The weak error estimates are based on
stochastic flows and discrete dual backward problems, and they can be used to
identify different error contributions arising from time and maturity
discretization as well as the classical statistical error due to finite
sampling. Explicit formulas for efficient computation of sharp error
approximation are included. Due to the structure of the HJM models considered
here, the computational effort devoted to the error estimates is low compared
to the work to compute Monte Carlo solutions to the HJM model. Numerical
examples with known exact solution are included in order to show the behavior
of the estimates
- …