This paper presents a graphical representation for the stochastic π-calculus, which is formalised by defining a corresponding graphical calculus. The graphical calculus is shown to be reduction equivalent to stochastic π, ensuring that the two calculi have the same expressive power. The graphical representation is used to model a couple of example biological systems, namely a bistable gene netw...

Proof. By composing Y with a homeomorphism R→ (0, 1) if required, we may assume that Y is bounded and we do so. To prove the claim, we appeal to the standard machinery using the monotone class theorem. Let L∞ be the family of bounded σ(X )-measurable functions and let H be the subset of L∞ defined by H = {Y ∈ L∞ : ∃ a Borel-measurable f satisfying Y = f (X ) a.s.}. We claim thatH = L∞. To this ...

Detereministic classical dynamics, reversible quantum dynamics in the Schrr odinger-Heinsenberg pictures and irreversible dynamics of diiusion processes driven by Brownian motion admitìnnnitesimal' or`diierential' descriptions which can be expressed in terms of derivations of * algebras. One of the central aims of quantum stochastic calculus is to unify all these and explore for similar algebra...

The issue of Quality of Service (QoS) performance analysis in packet switching networkshas drawn a lot of attention in the networking community in recent years. There is a lot ofwork including an elegant theory under the name of network calculus focusing on analysisof deterministic worst case QoS performance bounds. In the meantime, other researchershave studied the stochastic Q...

We develop a version of stochastic Pi-calculus endowed with a structural operational semantics expressed in terms of measure theory. The paper relies on two observations: (i) the structural congruence organizes a measurable space of processes and (ii) a well behaved SOS associates to each process, in a specified rate environment, a behaviour defined by a set of measures over the measurable spac...

These notes and other information about the course are available on www.statslab.cam.ac.uk/∼beresty/teach/StoCal/stocal.html Christina Goldschmidt, Stefan Grosskinsky, Gregory Miermont, and James Norris have all contributed to these notes. However, errors are my responsibility only. Comments and corrections are most welcome and should be sent to N.Berestycki AT statslab.cam.ac.uk Date last modi...

Preface This note contains a brief introduction to stochastic differential equations and their application in mathematical finance. It does not pretend to be complete in any respect. The topics of this note are covered in more detail in the course 04445 Statistics in Finance, and the interested reader may consult the adjacent lecture notes [8] or one of the standard texts [9, 4]. Throughout it ...

Cognitive radio has become a promising technology to increase spectrum utilization through spectrum sharing between licensed users (primary users) and unlicensed users (secondary users). Performance evaluation and analysis is of key importance to get more knowledge of this newly emerged technology. Queueing theory and Markov chain model have been applied to conduct the analysis, where some resu...