Nondifferentiable functions of one-dimensional semimartingales

We consider decompositions of processes of the form $Y=f(t,X_t)$ where $X$ is a semimartingale. The function $f$ is not required to be differentiable, so It\^{o}'s lemma does not apply. In the case where $f(t,x)$ is independent of $t$, it is shown that requiring $f$ to be locally Lipschitz continuous in $x$ is enough for an It\^{o}-style decomposition to exist. In particular, $Y$ will be a Dirichlet process. We also look at the case where $f(t,x)$ can depend on $t$, possibly discontinuously. It is shown, under some additional mild constraints on $f$, that the same decomposition still holds. Both these results follow as special cases of a more general decomposition which we prove, and which applies to nondifferentiable functions of Dirichlet processes. Possible applications of these results to the theory of one-dimensional diffusions are briefly discussed.Comment: Published in at http://dx.doi.org/10.1214/09-AOP476 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Dear Ocean

Poetry by Christine Lowther

Otolith by Emily Nilsen

Review of Emily Nilsen\u27s Otolith

Summertime Swamp Love by Patricia Young

Review of Patricia Young\u27s book of poems, Summertime Swamp Love

That Was Then, This Is Hell

Author describes a storm at her floating home and wonders it it is a by-product of human industry/ climate change

THEORETICAL IMPLICATIONS OF STRESS AND MAGNETIC-FIELD SPLITTING OF GR1-LINE IN DIAMOND

The g-value of the excited state associated with the GR1 line is analysed in terms of the Coulson-Kearsley defect-molecule model (1957) for the vacancy in diamond. It is shown that the experimentally observed g value is entirely consistent with the expected 1T2 level of the model provided quenching by the dynamic Jahn-Teller effect is considered. This quenching is estimated from stress splitting data for the GR1 line. The result supports strongly the view that the GR1 centre is the neutral vacancy

Design optimisation of electromagnetic devices using continuum design sensitivity analysis combined with commercial EM software

This paper deals with two types of optimisation problems: optimised source distribution and the shape optimum design, using Continuum Design Sensitivity Analysis (CDSA) in combination with standard electromagnetic (EM) software. Fast convergence and compatibility with existing EM software are the distinctive features of the proposed implementation. In order to verify the advantages and also to facilitate understanding of the method itself, two design optimisation problems have been tested using both 2D and 3D models: the first is a MRI design problem related to finding an optimal permanent magnet distribution and the second is a pole shape design problem to reduce the cogging torque in a BLDC

The implications of the use of composite materials in electromagnetic device topology and shape optimization

The use of soft magnetic composites (SMCs) provides electromagnetic device designers with serious challenges. While such materials have the potential of allowing novel 3-D topologies to be used, as well as providing major advantages in recycling at the end of the lifetime of the device, little or no design experience exists in this area. This paper proposes the use of sensitivity based topology optimization as a methodology for assisting designers in this task