Exponential martingales and changes of measure for counting processes

We give sufficient criteria for the Dol\'eans-Dade exponential of a stochastic integral with respect to a counting process local martingale to be a true martingale. The criteria are adapted particularly to the case of counting processes and are sufficiently weak to be useful and verifiable, as we illustrate by several examples. In particular, the criteria allow for the construction of for example nonexplosive Hawkes processes as well as counting processes with stochastic intensities depending on diffusion processes

Explicit formulae in probability and in statistical physics

We consider two aspects of Marc Yor's work that have had an impact in statistical physics: firstly, his results on the windings of planar Brownian motion and their implications for the study of polymers; secondly, his theory of exponential functionals of Levy processes and its connections with disordered systems. Particular emphasis is placed on techniques leading to explicit calculations.Comment: 14 pages, 2 figures. To appear in Seminaire de Probabilites, Special Issue Marc Yo

Supplementary Material for: Human Birth Weight and Reproductive Immunology: Testing for Interactions between Maternal and Offspring KIR and HLA-C Genes

<p><b><i>Background/Aims:</i></b> Maternal and offspring cell contact at the site of placentation presents a plausible setting for maternal-fetal genotype (MFG) interactions affecting fetal growth. We test hypotheses regarding killer cell immunoglobulin-like receptor <i>(KIR)</i> and <i>HLA-C</i> MFG effects on human birth weight by extending the quantitative MFG (QMFG) test. <b><i>Methods:</i></b> Until recently, association testing for MFG interactions had limited applications. To improve the ability to test for these interactions, we developed the extended QMFG test, a linear mixed-effect model that can use multi-locus genotype data from families. <b><i>Results:</i></b> We demonstrate the extended QMFG test's statistical properties. We also show that if an offspring-only model is fit when MFG effects exist, associations can be missed or misattributed. Furthermore, imprecisely modeling the effects of both <i>KIR</i> and <i>HLA-C</i> could result in a failure to replicate if these loci's allele frequencies differ among populations. To further illustrate the extended QMFG test's advantages, we apply the extended QMFG test to a UK cohort study and the Norwegian Mother and Child Cohort (MoBa) study. <b><i>Conclusion:</i></b> We find a significant <i>KIR</i>-<i>HLA-C</i> interaction effect on birth weight. More generally, the QMFG test can detect genetic associations that may be missed by standard genome-wide association studies for quantitative traits.</p

On continuity properties of the law of integrals of Lévy processes

Let $(\xi,\eta)$ be a bivariate L\'evy process such that the integral $\int\_0^\infty e^{-\xi\_{t-}} d\eta\_t$ converges almost surely. We characterise, in terms of their \LL measures, those L\'evy processes for which (the distribution of) this integral has atoms. We then turn attention to almost surely convergent integrals of the form $I:=\int\_0^\infty g(\xi\_t) dt$, where $g$ is a deterministic function. We give sufficient conditions ensuring that $I$ has no atoms, and under further conditions derive that $I$ has a Lebesgue density. The results are also extended to certain integrals of the form $\int\_0^\infty g(\xi\_t) dY\_t$, where $Y$ is an almost surely strictly increasing stochastic process, independent of $\xi$