A Hilbert space-valued stochastic integration is defined for an integrator that is a cylindrical fractional Brownian motion in a Hilbert space and an operator-valued integrand. Since the integrator is not a semimartingale for the fractional Brownian motions that are considered, a different definition of integration is required. Both deterministic and stochastic operator-valued integrands are used. The approach uses some ideas from Malliavin calculus. In addition to the definition of stochastic integration, an Itô formula is given for smooth functions of some processes that are obtained by the stochastic integration.

In this paper we examine the problem of existence and construction of multivariate Markov chains such that their components are Markov chains with given laws. Specifically, we provide sufficient and necessary conditions, in terms of semimartingale characteristics, for a component of a multivariate Markov chain to be a Markov chain in its own filtration -a property called weak Markov consistency. Accordingly, we introduce and discuss the concept of weak Markov copulae. Finally, we examine relationship between the concepts of weak Markov consistency and weak Markov copulae, and the corresponding strong versions of these concepts.

Lévy processes, Bond models, HJM postulate, Martingales, 60H30, 91B28, 60G51, E43, G12,