The paper presents the Stochastic Calculus of Looping Sequences (Stochastic CLS) suitable to describe microbiological systems and their evolution. The terms of the calculus are constructed by basic constituent elements and operators of sequencing, looping, containment and parallel composition. The looping operator allows tying up the ends of a sequence, thus creating a circular sequence which c...

Biological systems typically involve large numbers of components with complex, highly parallel interactions and intrinsic stochasticity. Numerous programming languages have been developed for modelling such systems, many of which are based on process calculi. Most of these calculi, particularly those involving membrane interactions, are expressive enough to generate potentially unbounded number...

We present an implementation of model checking for probabilistic and stochastic extensions of the π-calculus, a process algebra which supports modelling of concurrency and mobility. Formal verification techniques for such extensions have clear applications in several domains, including mobile ad-hoc network protocols, probabilistic security protocols and biological pathways. Despite this, no im...

This note is about Doob decomposition and the basics of Square integrable martingales Contents 1 Doob-Meyer Decomposition 1 2 Square Integrable Martingales 4 Brownian Motion and Stochastic Calculus Continuout Time Submartingales Usually it’s su¢ ce to only discuss submartingales by symmetry in de…nition and techniques are the same. 1 Doob-Meyer Decomposition Doob-meyer decomposition clears the ...

In this paper we propose a stochastic broadcast π-calculus which can be used to model server-client based systems where synchronization is always governed by only one participant. Therefore, there is no need to determine the joint synchronization rates. We also take immediate transitions into account which is useful to model behaviors with no impact on the temporal properties of a system. Since...

We provide a detailed hands-on tutorial for the R Development Core Team [2014] add-on package Sim.DiffProc [Guidoum and Boukhetala, 2014], for symbolic and floating point computations in stochastic calculus and stochastic differential equations (SDEs). The package implement is introduced and it is explains how to use the snssde1d, snssde2d and snssde3d main functions in this package, for simula...

– P is the “true” or physical probability measure – Ω is the universe of possible outcomes. We use ω ∈ Ω to represent a generic outcome, typically a sample path(s) of a stochastic process(es). – the set F represents the set of possible events where an event is a subset of Ω. • There is also a filtration, {F}t≥0, that models the evolution of information through time. So for example, if it is kno...

We present an implementation of model checking for probabilistic and stochastic extensions of the -calculus, a process algebra which supports modeling of concurrency and mobility. Formal verification techniques for such extensions have clear applications in several domains, including mobile ad hoc network protocols, probabilistic security protocols, and biological pathways. Despite this, no imp...

Abstract: We study densely defined unbounded operators acting between different Hilbert spaces. For these, we introduce a notion of symmetric (closable) pairs of operators. The purpose of our paper is to give applications to selected themes at the cross road of operator commutation relations and stochastic calculus. We study a family of representations of the canonical commutation relations (CC...

During the last two decades, starting with the seminal work by Cruz, network calculus has evolved as a new theory for the performance analysis of networked systems. In contrast to classical queueing theory, it deals with performance bounds instead of average values and thus has been the theoretical basis of quality of service proposals such as the IETF’s Integrated andDifferentiated Services ar...