Ely Merzbach

STOPPING FOR TWO-DIMENSIONAL STOCHASTIC PROCESSES

The aim of this paper is to introduce some techniques that can be used in the study of stochastic processes which have as parameter set the positive quadrant of the plane R2+. We define stopping lines and derive an interesting property of measurability for them. The notion of predictability is developed, and we show the connection between predictable processes, fields associated with stopping lines, and predictable stopping lines. We also give a theorem of section for predictable sets. Extension to processes indexed by any partially ordered set with some regularity assumptions can be carried out quite easily with the same techniques.

What is a multi-parameter renewal process?

The concept of the renewal property is extended to processes indexed by a multidimensional time parameter. The definition given includes not only partial sum processes, but also Poisson processes and many other point processes whose jump points are not totally ordered. A new version of the waiting time paradox is proven for multidimensional Poisson processes, and is shown to imply the renewal property. Finally, martingale properties of renewal processes are studied.

A note on expansion for functionals of spatial marked point processes

Expansion of the mean value of a functional of a spatial marked point process with respect to the factorial moment measures is presented. This paper complements previous studies of a point process on the real line, by extending the results to a general Polish space.

Stopping and set-indexed local martingales

Set-indexed local martingales are defined and studied. We present some optional sampling theorems for strong martingales, martingales and weak martingales. The class of set-indexed processes which are locally of class (D) is introduced. A Doob-Meyer decomposition is obtained: any local weak submartingale has a unique decomposition into the sum of a local weak martingale and a local predictable increasing process. Finally some examples are given.

The Multiparameter Fractional Brownian Motion

We define and study the multiparameter fractional Brownian motion. This process is a generalization of both the classical fractional Brownian motion and the multiparameter Brownian motion, when the condition of independence is relaxed. Relations with the Levy fractional Brownian motion and with the fractional Brownian sheet are discussed. Different notions of stationarity of the increments for a multiparameter process are studied and applied to the fractional property. Using self-similarity we present a characterization for such processes. Finally, behavior of the multiparameter fractional Brownian motion along increasing paths is analysed.

A martingale characterization of the set-indexed poisson process

A martingale characterization of the set-indexed Poisson process is proved: a set-indexed point process is a Poisson process if and only if there exists a deterministic and increasing process such that the difference is a strong martingale

Predictability and stopping on lattices of sets

SummaryAs a first step in the development of a general theory of set-indexed martingales, we define predictability on a general space with respect to a filtration indexed by a lattice of sets. We prove a characterization of the predictable σ-algebra in terms of adapted and “left-continuous” processes without any form of topology for the index set. We then define a stopping set and show that it is a natural generalization of the stopping time; in particular, the predictable σ-algebra can be characterized by various stochastic intervals generated by stopping sets.

On the extension of bimeasures

We prove a necessary and sufficient condition for the existence of an extension of a scalar bimeasure on abstract sets to a Σ-additive measure on the generated Σ-algebra. We also prove some extension theorems for vector bimeasures.

Point processes indexed by directed sets

The concept of point process is defined where the parameter set is a directed set. By extending the parameter set, every point process can be described as an integer measure on a well-ordered set, and then the predictable projection of a point process can be constructed.

Different kinds of two-parameter martingales

This paper is devoted to the study of all the different classes of two-parameter martingales which were introduced during the last decade. The problem of the relations between these classes is completely solved under the F-4 assumption and counter-examples are given in order to point out the differences between these classes. Characterizations in terms of Doob-Meyer-Cairoli decompositions are obtained. In the case where the filtration is generated by a two-parameter Poisson process, we prove that the class of strong martingales coincides with the class of martingales of direction independent variation.