Stochastic completeness and volume growth

Lack of strong completeness for stochastic flows

It is well-known that a stochastic differential equation (SDE) on a Euclidean space driven by a Brownian motion with Lipschitz coefficients generates a stochastic flow of homeomorphisms. When the coefficients are only locally Lipschitz, then a maximal continuous flow still exists but explosion in finite time may occur. If – in addition – the coefficients grow at most linearly, then this flow ha...

Dirichlet Forms and Stochastic Completeness of Graphs and Subgraphs

We characterize stochastic completeness for regular Dirichlet forms on discrete sets. We then study how stochastic completeness of a subgraph is related to stochastic completeness of the whole graph. We show that any graph is a subgraph of a stochastically complete graph and that stochastic incompleteness of a suitably modified subgraph implies stochastic incompleteness of the whole graph. Alon...

Contingent Claims and Market Completeness in a Stochastic Volatility Model

In an incomplete market framework, contingent claims are of particular interest since they improve the market efficiency. This paper addresses the problem of market completeness when trading in contingent claims is allowed. We extend recent results by Bajeux and Rochet (1996) in a stochastic volatility model to the case where the asset price and its volatility variations are correlated. We also...

Completeness and quasi-completeness

The appropriate general completeness notion for topological vector spaces is quasi-completeness. There is a stronger general notion of completeness, which proves to be too strong in general. For example, the appendix shows that weak-star duals of infinite-dimensional Hilbert spaces are quasi-complete, but never complete in the stronger sense. Quasi-completeness for Fréchet spaces is ordinary me...

Stochastic models , estimation , and control VOLUME