Motivated by the well-posedness of birth-and-growth processes, a stochastic geometric differential equation and, hence, a stochastic geometric dynamical system are proposed. In fact, a birth-and-growth process can be rigorously modeled as a suitable combination, involving the Minkowski sum and the Aumann integral, of two very general set-valued processes representing nucleation and growth dynamics, respectively. The simplicity of the proposed geometric approach allows to avoid problems of boundary regularities arising from an analytical definition of the front growth. In this framework, growth is generally anisotropic and, according to a mesoscale point of view, is non local, i.e. at a fixed time instant, growth is the same at each point of the space.

Multitype branching processes and Feller diffusion processes are conditioned on very late extinction. The conditioned laws are expressed as Doob h-transforms of the unconditioned laws, and an interpretation of the conditioned paths for the branching process is given, via the immortal particle. We study different limits for the conditioned process (increasing delay of extinction, long-time behavior, scaling limit) and provide an exhaustive list of exchangeability results.

On donne des exemples d'entrelacements entre semi-groupes markoviens obtenus au moyen de considérations de théorie des groupes sur les paires de Gelfand

We solve explicitly the following problem: for a given probability measure μ, we specify a generalised martingale diffusion (Xt) which, stopped at an independent exponential time T, is distributed according to μ. The process (Xt) is specified via its speed measure m. We present two heuristic arguments and three proofs. First we show how the result can be derived from the solution of [Bertoin and Le Jan, Ann. Probab. 20 (1992) 538–548.] to the Skorokhod embedding problem. Secondly, we give a proof exploiting applications of Krein's spectral theory of strings to the study of linear diffusions. Finally, we present a novel direct probabilistic proof based on a coupling argument.

A general theory is developed for the projection of martingale related processes onto smaller filtrations, to which they are not even adapted. Martingales, supermartingales, and semimartingales retain their nature, but the case of local martingales is more delicate, as illustrated by an explicit case study for the inverse Bessel process. This has implications for the concept of No Free Lunch with Vanishing Risk, in Finance.

Let (Ω, $\mathcal{F}$, ($\mathcal{F}_t$)t≥0, $\mathbb{P}$) be a filtered probability space satisfying the usual assumptions: it is usually not possible to extend to $\mathcal{F}_{\infty}$ (the σ-algebra generated by ($\mathcal{F}_t$)t≥0) a coherent family of probability measures ($\mathbb{Q}_t$) indexed by t ≥ 0, eachof them being defined on $\mathcal{F}_t$. It is known that for instance, on the Wiener space, this extension problem has a positive answer if one takes the filtration generated by the coordinate process, made right-continuous, but can have a negative answer if one takes its usual augmentation. On the other hand, the usual assumptions are crucial in order to obtain the existence of regular versions of paths (typically adapted and continuous or adapted and càdlàg versions) for most stochastic processes of interest, such as the local time of the standard Brownian motion, stochastic integrals, etc. For instance we shall prove that on the Wiener space, equipped with the right-continuous augmentation of the filtration generated by the canonical process, there exists no càdlàg or continuous and adapted version for the local time at level zero of the canonical process. Hence there is an incompatibility between the problem of extending a coherent family of probability measures to $\mathcal{F}_\infty$ and the classical construction of regular versions of paths, requiring the usual assumptions (this situation typically occurs in the problem of penalization of the Brownian paths or in mathematical finance), which to the best of our knowledge, has not been noticed so far.In order to fix this problem, we introduce a new property for filtrations, intermediate between the right continuityand the usual conditions. More precisely, we say that a filtration ($\mathcal{F}_t$)t≥0 satisfies the natural assumptionsif it is right-continuous and if $\mathcal{F}_0$ contains all the sets included in a countable unionof negligible sets (Bn)n≥1, such that Bn ∈ $\mathcal{F}_n$ for n ≥ 1.There is a natural way to obtain, from a given filtration ($\mathcal{F}_t$)t≥0, a new filtration which satisfies the natural assumptions: we call it the natural augmentation of ($\mathcal{F}_t$)t≥0.We show that most of the important results of the theory of stochastic processes which are generally proved under the usual augmentation, such as the existence of regular version of trajectories or the début theorem, still hold under the natural augmentation; moreover this new augmentation allows the extension of a coherent family of probability measures whenever this is possible with the original filtration. For sake of completeness, we also recall (not so well known) Parthasarathy type conditions on the underlying filtration under which the extension problem for a coherent family of probability measures has a solution. In particular, we shall see that this is always the case on the following two fundamental spaces: $\mathcal{C}$($\mathbb{R}_+$, $\mathbb{R}$), the space of continuous functions equipped with the filtration generated by the coordinate process and $\mathcal{D}$($\mathbb{R}_+$, $\mathbb{R}$), the space of càdlàg functions endowed with the filtration generated by the coordinate process.

It is proved that each Hoeffding space associated with a random permutation(or, equivalently, with extractions without replacement from a finitepopulation) carries an irreducible representation of the symmetric group,equivalent to a two-block Specht module.

Wiener integral for the coordinate process is defined under the σ-finite measure unifying Brownian penalisations, which has been introduced by [Najnudel et al., C. R. Math. Acad. Sci. Paris 345 (2007) 459–466] and [Najnudel et al., MSJ Memoirs 19. Mathematical Society of Japan, Tokyo (2009)]. Its decomposition before and after last exit time from 0 is studied. This study prepares for the author's recent study [K. Yano, J. Funct. Anal. 258 (2010) 3492–3516] of Cameron-Martin formula for the σ-finite measure.