Maximal Inequalities for Fractional Lévy and Related Processes

In this article we study processes that are constructed by a convolution of a deterministic kernel with a martingale. A special emphasis is put on the case where the driving martingale is a centred Lévy process, which covers the popular class of fractional Lévy processes. As a main result we show that, under appropriate assumptions on the kernel and the martingale, the maximum process of the corresponding “convoluted martingale” is p-integrable and we derive maximal inequalities in terms of the kernel and of the moments of the driving martingale.     

The proportional hazards regression model with staggered entries: A strong martingale approach

The proportional hazards regression model, when subjects enter the study in a staggered fashion, is studied. A strong martingale approach is used to model the two-time parameter counting processes. It is shown that well-known univariate results such as weak convergence and martingale inequalities can be extended to this two-dimensional model. Strong martingale theory is also used to prove weight convergence of a general weighted goodness-of-fit process and its weighted bootstrap counterpart.     

Coupling and option price comparisons in a jump-diffusion model

In this paper, we examine the dependence of option prices in a general jump-diffusion model on the choice of martingale pricing measure. Since the model is incomplete, there are many equivalent martingale measures. Each of these measures corresponds to a choice for the market price of diffusion risk and the market price of jump risk. Our main result is to show that for convex payoffs, the option price is increasing in the jump-risk parameter. We apply this result to deduce general inequalities, comparing the prices of contingent claims under various martingale measures, which have been proposed in the literature as candidate pricing measures.

Our proofs are based on couplings of stochastic processes. If there is only one possible jump size then we are able to utilize a second coupling to extend our results to include stochastic jump intensities.     

Common strict character of some sharp infinite-sequence martingale inequalities

A procedure is given for proving strictness of some sharp, infinite-sequence martingale inequalities, which arise from sharp, finite-sequence martingale inequalities attained by degenerating extremal distributions. The procedure is applied to obtain strictness of the sharp inequalities of Cox and Kemperman

$\text{P(|X}{}_{\mathrm{i}}\text{|?1 for some i=1, 2,…)?(ln 2)}{}^{\mathrm{?1}}\text{sup}\text{n}\text{E}\text{∑}\text{i=0}\text{n}\text{X}{}_{\mathrm{i}}$

and of Cox (sharp form of Burkholder's inequality)

$\text{P}\text{∑}\text{i=0}\text{∞}\text{X}{}^2{}_{\mathrm{i}}\text{?1}\text{? e}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{sup}\text{n}\text{E}\text{∑}\text{i=0}\text{n}\text{X}{}_{\mathrm{i}}$

for all nontrivial martingale difference sequences X₀,X₁,….     

The joint law of the maximum and terminal value of a martingale

Summary In this paper, we characterise the possible joint laws of the maximum and terminal value of a uniformly-integrable martingale. We also characterise the joint laws of the maximum and terminal value of a convergent continuous local martingale vanishing at zero. A number of earlier results on the possible laws of the maximum can be deduced quite easily.     

A probabilistic approach to Martin boundaries for manifolds with ends

Summary We use martingale methods and coupling arguments to prove results of Li and Tam (1987) and Donnelly (1986) characterizing positive and bounded harmonic functions, respectively, on certain manifolds with ends.Research supported by a grant from NSA/NSF     

Central Limit Theorems for approximate quadratic variations of pure jump Itô semimartingales

We derive Central Limit Theorems for the convergence of approximate quadratic variations, computed on the basis of regularly spaced observation times of the underlying process, toward the true quadratic variation. This problem was solved in the case of an Itô semimartingale having a non-vanishing continuous martingale part. Here we focus on the case where the continuous martingale part vanishes and find faster rates of convergence, as well as very different limiting processes.     

Point processes in the plane

In this survey paper, two-parameter point processes are studied in connection with martingale theory and with respect to the partial-order induced by the Cartesian coordinates of the plane. Point processes are characterized by jump stopping times and by their two-parameter compensators. Properties of the doubly stochastic Poisson process, such as predictability, are discussed. A definition for the Palm measure of a two-parameter stationary point process is proposed.     

Burkholder-Gundy-Davis inequality in martingale Hardy spaces with variable exponent

In this article, by extending classical Dellacherie's theorem on stochastic sequences to variable exponent spaces, we prove that the famous Burkholder-Gundy-Davis inequality holds for martingales in variable exponent Hardy spaces. We also obtain the variable exponent analogues of several martingale inequalities in classical theory, including convexity lemma, Chevalier's inequality and the equivalence of two kinds of martingale spaces with predictable control. Moreover, under the regular condition on σ-algebra sequence we prove the equivalence between five kinds of variable exponent martingale Hardy spaces.     

Comparing the minimal Hellinger martingale measure of order q to the q-optimal martingale measure

This paper investigates the relationship between the minimal Hellinger martingale measure of order q$q$ (MHM measure hereafter) and the q$q$-optimal martingale measure for any q≠1$q\ne 1$. First, we provide more results for the MHM measure; in particular we establish its complete characterization in two manners. Then we derive two equivalent conditions for both martingale measures to coincide. These conditions are in particular fulfilled in the case of markets driven by Lévy processes. Finally, we analyze the MHM measure as well as its relationship to the q$q$-optimal martingale measure for the case of a discrete-time market model.     

Law of the iterated logarithm for the periodogram

We consider the almost sure asymptotic behavior of the periodogram of stationary and ergodic sequences. Under mild conditions we establish that the limsup of the periodogram properly normalized identifies almost surely the spectral density function associated with the stationary process. Results for a specified frequency are also given. Our results also lead to the law of the iterated logarithm for the real and imaginary parts of the discrete Fourier transform. The proofs rely on martingale approximations combined with results from harmonic analysis and techniques from ergodic theory. Several applications to linear processes and their functionals, iterated random functions, mixing structures and Markov chains are also presented.     

Martingale solutions and Markov selection of stochastic 3D Navier-Stokes equations with jump

In this paper, we study the existence of martingale solutions of stochastic 3D Navier-Stokes equations with jump, and following Flandoli and Romito (2008) [7] and Goldys et al. (2009) [8], we prove the existence of Markov selections for the martingale solutions.     

The importance of strictly local martingales; applications to radial Ornstein–Uhlenbeck processes

For a wide class of local martingales (M _t ) there is a default function, which is not identically zero only when (M _t ) is strictly local, i.e. not a true martingale. This default in the martingale property allows us to characterize the integrability of functions of sup _s≤t M _s in terms of the integrability of the function itself. We describe some (paradoxical) mean-decreasing local sub-martingales, and the default functions for Bessel processes and radial Ornstein–Uhlenbeck processes in relation to their first hitting and last exit times. Received: 6 August 1996 / Revised version: 27 July 1998     

Enlargement of the Wiener filtration by an absolutely continuous random variable via Malliavin’s calculus

Summary. The analytic treatment of problems related to the asymptotic behaviour of random dynamical systems generated by stochastic differential equations suffers from the presence of non-adapted random invariant measures. Semimartingale theory becomes accessible if the underlying Wiener filtration is enlarged by the information carried by the orthogonal projectors on the Oseledets spaces of the (linearized) system. We study the corresponding problem of preservation of the semimartingale property and the validity of a priori inequalities between the norms of stochastic integrals in the enlarged filtration and norms of their quadratic variations in case the random element F enlarging the filtration is real valued and possesses an absolutely continuous law. Applying the tools of Malliavin’s calculus, we give smoothness conditions on F under which the semimartingale property is preserved and a priori martingale inequalities are valid. Received: 12 April 1995 / In revised form: 7 March 1996     

Embedding and asymptotic expansions for martingales

The paper develops a way of embedding general martingales in continuous ones in such a way that the quadratic variation of the continuous martingale has conditional cumulants (given the original martingale) that are explicitly given in terms of optional and predictable variations of the original process. Bartlett identities for the conditional cumulants are also found. A main corollary to these results is the establishment of second (and in some cases higher) order asymptotic expansions for martingales.Research supported in part by National Science Foundation grant DMS 93-05601 and Army Research Office grant DAAH04-1-0105     

Uniqueness and absolute continuity of weak solutions for parabolic SPDE's

The paper presents necessary and sufficient conditions for the absolute continuity of measures generated by infinite-dimensional martingale problems. This result is applied to the study of the existence and uniqueness of weak solutions to nonlinear parabolic SPDE's. The paper also addresses the problem of stochastic integration with respect to a martingale in a quasi-complete locally convex topological vector space.This work was partially supported by NSF Grant # DMS-9002997 and ONR Grant # N00014-91-J-1526.     

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.     

Information,no-arbitrage and completeness for asset price models with a change point

We consider a general class of continuous asset price models where the drift and the volatility functions, as well as the driving Brownian motions, change at a random time τ$\tau$. Under minimal assumptions on the random time and on the driving Brownian motions, we study the behavior of the model in all the filtrations which naturally arise in this setting, establishing martingale representation results and characterizing the validity of the NA1 and NFLVR no-arbitrage conditions.     

Regularity of Skorohod integral processes based on integrands in a finite Wiener chaos

Summary Letf be a square integrable kernel on them-dimensional unit cube,U the Skorohod integral process in them ^th Wiener chaos associated with it. Isoperimetric inequalities for functions on Wiener space yield the exponential integrability of the increments ofU. To this result we apply the majorizing measure technique to show thatU possesses a continuous version and give an upper bound of its modulus of continuity.     

Representations of the optimal filter in the context of nonlinear filtering of random fields with fractional noise

The problem of nonlinear filtering of multiparameter random fields, observed in the presence of a long-range dependent spatial noise, is considered. When the observation noise is modelled by a persistent fractional Wiener sheet, several pathwise representations of the optimal filter are derived. The representations involve series of multiple stochastic integrals of different types and are particularly important since the evolution equations, satisfied by the best mean-square estimate of the signal random field, have a complicated analytical structure and fail to be proper (measure-valued) stochastic partial differential equations. Several of the above optimal filter representations involve a new family of strong martingale transforms associated to the multiparameter fractional Brownian sheet; the latter martingale family is of independent interest in fractional stochastic calculus of multiparameter random fields.