On the characterisation of honest times that avoid all stopping times

We present a short and self-contained proof of the following result: a random time is an honest time that avoids all stopping times if and only if it coincides with the (last) time of maximum of a nonnegative local martingale with zero terminal value and no jumps while at its running supremum, where the latter running supremum process is continuous. Illustrative examples involving local martingales with discontinuous paths are provided.     

On tightness and stopping times

This paper studies conditions of tightness for sequences of processes, which conditions are mostly based on the use of ‘dominating’ increasing processes. The results obtained follow in directions initiated by Aldous and Rebolledo and are particularly well-suited for studying sequences of semimartingales. Also obtained are results that extend sufficient conditions of Aldous's type to processes that are not quasi-left-continuous.     

Selector processes on classes of sets

Consider the random subset X of ℕ obtained by selecting independently each integer with a probability δ. Consider a finite class of finite sets. We describe a combinatorial quantity that is of the same order as We then give a related result allowing to compute the supremum of the empirical process on a class of sets. Work partially supported by an NSF grant.     

Optimal stopping with irregular reward functions

We consider optimal stopping problems with finite horizon for one-dimensional diffusions. We assume that the reward function is bounded and Borel-measurable, and we prove that the value function is continuous and can be characterized as the unique solution of a variational inequality in the sense of distributions.     

Geography of the level sets of the Brownian sheet

Summary We describe geometric properties of {W>}, whereW is a standard real-valued Brownian sheet, in the neighborhood of the first hitP of the level set {W>} along a straight line or smooth monotone curveL. In such a neighborhood we use a decomposition of the formW(s, t)=–b(s)+B(t)+x(s, t), whereb(s) andB(t) are particular diffusion processes andx(s, t) is comparatively small, to show thatP is not on the boundary of any connected component of {W>}. Rather, components of this set form clusters nearP. An integral test for thorn-shaped neighborhoods ofL with tip atP that do not meet {W>} is given. We then analyse the position and size of clusters and individual connected components of {W>} near such a thorn, giving upper bounds on their height, width and the space between clusters. This provides a local picture of the level set. Our calculations are based on estimates of the length of excursions ofB andb and an accounting of the error termx.The research of this author was partially supported by NSF grant DMS-9103962, and, during the period of revision, by grant DAAL03-92-6-0323 from the Army Research Office     

Applications of the degree theorem to absolute continuity on Wiener space

Summary Let (,H, P) be an abstract Wiener space and define a shift on byT()=+F() whereF is anH-valued random variable. We study the absolute continuity of the measuresPºT ^–1and ( _F P)ºT ¹ with respect toP using the techniques of the degree theory of Wiener maps, where _F =det₂(1+F) × Exp{–F–1/2|F|²}.The work of the second author was supported by the fund for promotion of research at the Technion     

A short proof of the Doob–Meyer theorem

Every submartingale S$S$ of class D$D$ has a unique Doob–Meyer decomposition S=M+A$S=M+A$, where M$M$ is a martingale and A$A$ is a predictable increasing process starting at 0.     

Kernel B-splines on general lattices

The aim of the paper is to provide a computationally effective way to construct stable bases on general non-degenerate lattices. In particular, we define new stable bases on hexagonal lattices and we give some numerical examples which show their usefulness in applications.     

Selfdecomposability of moving average fractional Lévy processes

We study the relationships between the selfdecomposability of marginal distributions or finite dimensional distributions of moving average fractional Lévy processes and distributions of their driving Lévy processes.     

Characterization of infinite divisibility by duality formulas. Application to Lévy processes and random measures

Processes with independent increments are proven to be the unique solutions of duality formulas. This result is based on a simple characterization of infinitely divisible random vectors by a functional equation in which a difference operator appears. This operator is constructed by a variational method and compared to approaches involving chaos decompositions. We also obtain a related characterization of infinitely divisible random measures.     

Tempered stable distributions and processes

We investigate the class of tempered stable distributions and their associated processes. Our analysis of tempered stable distributions includes limit distributions, parameter estimation and the study of their densities. Regarding tempered stable processes, we deal with density transformations and compute their p$p$-variation indices. Exponential stock models driven by tempered stable processes are discussed as well.     

A simulation approach to optimal stopping under partial information

We study the numerical solution of nonlinear partially observed optimal stopping problems. The system state is taken to be a multi-dimensional diffusion and drives the drift of the observation process, which is another multi-dimensional diffusion with correlated noise. Such models where the controller is not fully aware of her environment are of interest in applied probability and financial mathematics. We propose a new approximate numerical algorithm based on the particle filtering and regression Monte Carlo methods. The algorithm maintains a continuous state space and yields an integrated approach to the filtering and control sub-problems. Our approach is entirely simulation-based and therefore allows for a robust implementation with respect to model specification. We carry out the error analysis of our scheme and illustrate with several computational examples. An extension to discretely observed stochastic volatility models is also considered.     

Birth delay of a Markov process and the stopping distributions for regular processes

Summary We consider Markov processes with a fixed transition functionp(r, x; t, B) and with random birth times. We show that a process can be obtained from (X _t ,P) by birth delay if and only if for allt andB. As an application, we give a new version and a new proof of the results of Rost [R] and Fitzsimmons [F2] on stopping distributions of Markov processes. The key Lemma 1.1 replaces the filling scheme used by the previous authors.Birth delay was considered from a different prospective in [F1].Partially supported by the National Science Foundation Grant DMS-8802667     

Factorial moments of point processes

We derive joint factorial moment identities for point processes with Papangelou intensities. Our proof simplifies previous combinatorial approaches to the computation of moments for point processes. We also obtain new explicit sufficient conditions for the distributional invariance of point processes with Papangelou intensities under random transformations.     

Random rotations of the Wiener path

Summary Let (W, H, ) be an abstract Wiener space and letR(w) be a strongly measurable random variable with values in the set of isometries onH. Suppose that Rh is smooth in the Sobolev sense and that it is a quasi-nilpotent operator onH for everyhH. It is shown that (R(w)h) is again a Gaussian (0, |h| _H ² )-random variable. Consequently, if (e _i ,i)W ^* is a complete, orthonormal basis ofH, then defines a measure preserving transformation, a rotation, onW. It is also shown that if for some strongly measurable, operator valued (onH) random variableR, (R(w+k)h) is (0, |h| _H ² )-Gaussian for allk, hH, thenR is an isometry and Rh is quasi-nilpotent for allHH. The relation between the stochastic calculi for these Wiener pathsw and , as well as the conditions of the inverbibility of the map are discussed and the problem of the absolute continuity of the image of the Wiener measure under Euclidean motion on the Wiener space (i.e. composed with a shift) is studied.The research of the second author was supported by the Fund for the Promotion of Research at the TechnionDedicated to the memory of Albert Badrikian     

Green function estimates for relativistic stable processes in half-space-like open sets

In this paper, we establish sharp two-sided estimates for the Green functions of relativistic stable processes (i.e. Green functions for non-local operators m−(m^2/α−Δ)^α/2) in half-space-like C^1,1 open sets. The estimates are uniform in m∈(0,M] for each fixed M∈(0,∞). When m↓0, our estimates reduce to the sharp Green function estimates for −(−Δ)^α/2 in such kind of open sets that were obtained recently in Chen and Tokle [12]. As a tool for proving our Green function estimates, we show that a boundary Harnack principle for X^m, which is uniform for all m∈(0,∞), holds for a large class of non-smooth open sets.     

Law of large numbers and central limit theorem for Donsker's delta function of diffusions I

Limit theorems in the space of Hida distributions, similar to the law of large numbers and the central limit theorem, are shown for composites of the Dirac distribution with solutions of one-dimensional, non-degenerate Itô equations.Supported by National Science Foundation under grant DMS-9001859.Supported by the Louisiana Education Quality Support Fund under grant (91–93) RD-A-08.Supported by the Council on Research of Louisiana State University.     

Finite approximation schemes for Lévy processes,and their application to optimal stopping problems

This paper proposes two related approximation schemes, based on a discrete grid on a finite time interval [0,T]$[0,T]$, and having a finite number of states, for a pure jump Lévy process _Lt$L_t$. The sequences of discrete processes converge to the original process, as the time interval becomes finer and the number of states grows larger, in various modes of weak and strong convergence, according to the way they are constructed. An important feature is that the filtrations generated at each stage by the approximations are sub-filtrations of the filtration generated by the continuous time Lévy process. This property is useful for applications of these results, especially to optimal stopping problems, as we illustrate with an application to American option pricing. The rates of convergence of the discrete approximations to the underlying continuous time process are assessed in terms of a “complexity” measure for the option pricing algorithm.     

A central limit theorem for stationary random fields

This paper establishes a central limit theorem and an invariance principle for a wide class of stationary random fields under natural and easily verifiable conditions. More precisely, we deal with random fields of the form _Xk=g(_εk−s,s∈Z^d)$X_k=g({\epsilon }_{k-s},s\in {\mathbb{Z}}^d)$, k∈Z^d$k\in {\mathbb{Z}}^d$, where _(εi)i∈Zd${\left({\epsilon }_i\right)}_{i\in {\mathbb{Z}}^d}$ are iid random variables and g$g$ is a measurable function. Such kind of spatial processes provides a general framework for stationary ergodic random fields. Under a short-range dependence condition, we show that the central limit theorem holds without any assumption on the underlying domain on which the process is observed. A limit theorem for the sample auto-covariance function is also established.     

Moment formulae for general point processes

The goal of this paper is to generalize most of the moment formulae obtained in [12]. More precisely, we consider a general point process μ, and show that the quantities relevant to our problem are the so-called Papangelou intensities. When the Papangelou intensities of μ are well-defined, we show some general formulae to recover the moment of order n of the stochastic integral of the point process. We will use these extended results to introduce a divergence operator and study a random transformation of the point process.