Gaussian density estimates for the solution of singular stochastic Riccati equations

summary:Stochastic Riccati equation is a backward stochastic differential equation with singular generator which arises naturally in the study of stochastic linear-quadratic optimal control problems. In this paper, we obtain Gaussian density estimates for the solutions to this equation

Statistical analysis of quantum-entangled-network generation

We develop techniques to analyze the statistics of completion times of nondeterministic elements in quantum-entanglement generation and how they affect the overall performance as measured by the secret key rate. By considering such processes as Markov chains, we show how to obtain exact expressions for the probability distributions over the number of errors that a network acquires, as well as the distribution of entanglement establishment times. We show how results from complex analysis can be used to analyze Markov matrices to extract information with a lower computational complexity than previous methods. We apply these techniques to the Briegel et al. quantum repeater protocol [H.-J. Briegel et al., Phys. Rev. Lett. 81, 5932 (1998)] and find that consideration of the effect of statistical fluctuations can tighten bounds on the secret key rate by three orders of magnitude, when compared to more simplistic analyses. We also use the theory of order statistics to derive tighter bounds on the minimum quantum memory lifetimes that are required in order to communicate securely

Anticipating integrals and martingales on the Poisson space

Let Ñt be a standard compensated Poisson process on [0, 1]. We prove a new characterization of anticipating integrals of the Skorohod type with respect to Ñ, and use it to obtain several counterparts to well established properties of semimartingale stochastic integrals. In particular we show that, if the integrand is sufficiently regular, anticipating Skorohod integral processes with respect to Ñ admit a pointwise representation as usual Itô integrals in an independently enlarged filtration. We apply such a result to: (i) characterize Skorohod integral processes in terms of products of backward and forward Poisson martingales, (ii) develop a new Itôtype calculus for anticipating integrals on the Poisson space, and (iii) write Burkholder-type inequalities for Skorohod integrals

Chaotic Kabanov formula for the Azéma martingales

We derive the chaotic expansion of the product of n-th and first order multiple stochastic integrals with respect to certain normal martingales. This is done by application of the classical and quantum product formulas for multiple stochastic integrals. Our approach extends existing results on chaotic calculus for normal martingales and exhibits properties relative to multiple stochastic integrals, polynomials and Wick products, that characterize the Wiener and Poisson processes. Key words: Az'ema martingales, multiple stochastic integrals, product formulas. Mathematics Subject Classification (1991): 60G44, 60H05, 81S25. 1 Introduction The Wiener-Ito and Poisson-Ito chaotic decompositions give an isometric isomorphism between the Fock space \Gamma(L 2 (IR + )) and the space of square-integrable functionals of the process. This somorphism is constructed by association of a symmetric function f n 2 L 2 (IR + ) ffin to its multiple stochastic integral. The isometry property comes ..