AimLately, the diagnostic value of magnetic resonance imaging, Lasègue sign and classic neurological signs have been considered not accurate enough to distinguish the radicular from non-radicular low back with leg pain (LBLP) and a calculation of the symptomatic side muscle volume has been indicated as a probable valuable marker. However, only the multifidus muscle volume has been calculated so far. The main objective of the study was to verify whether LBLP subjects presented symptomatic side pelvic muscle atrophy compared to healthy volunteers. The second aim was to assess the inter-rater reliability of 3-D manual method for segmenting and measuring the volume of the gluteus maximus, gluteus medius, gluteus minimus and piriformis muscles in both LBLP patients and healthy subjects.MethodTwo independent raters analyzed MR images of LBLP and healthy subjects towards muscle volume of four pelvic muscles, i.e. the piriformis, gluteus minimus, gluteus medius and gluteus maximus. For both sides, the MR images of the muscles without adipose tissue infiltration were manually segmented in 3-D medical images.ResultsSymptomatic muscle atrophy was confirmed in only over 50% of LBLP subjects (gluteus maximus (p<0.001), gluteus minimus (p<0.01) and piriformis (p<0.05)). The ICC values indicated that the inter-rater reproducibility was greater than 0.90 for all measurements (LBLP and healthy subjects), except for the measurement of the right gluteus medius muscle in LBLP patients, which was equal to 0.848.ConclusionMore than 50% of LBLP subjects presented symptomatic gluteus maximus, gluteus minimus and piriformis muscle atrophy. 3-D manual segmentation reliably measured muscle volume in all the measured pelvic muscles in both healthy and LBLP subjects. To answer the question of what kind of muscle atrophy is indicative of radicular or non-radicular pain further studies are required.

The truncated variation, TV c , is a fairly new concept introduced in [5]. Roughly speaking, given a càdlàg function f , its truncated variation is "the total variation which does not pay attention to small changes of f , below some threshold c > 0". The very basic consequence of such approach is that contrary to the total variation, TV c is always finite. This is appealing to the stochastic analysis where so-far large classes of processes, like semimartingales or diffusions, could not be studied with the total variation. Recently in [6], another characterization of TV c was found. Namely TV c is the smallest possible total variation of a function which approximates f uniformly with accuracy c/2. Due to these properties we envisage that TV c might be a useful concept both in the theory and applications of stochastic processes.For this reason we decided to determine some properties of TV c for some well-known processes. In course of our research we discover intimate connections with already known concepts of the stochastic processes theory.Firstly, for semimartingales we proved that TV c is of order c −1 and the normalized truncated variation converges almost surely to the quadratic variation of the semimartingale as c ց 0. Secondly, we studied the rate of this convergence. As this task was much more demanding we narrowed to the class of diffusions (with some mild additional assumptions). We obtained the weak convergence to a so-called Ocone martingale. These results can be viewed as some kind of law of large numbers and the corresponding central limit theorem.Finally, for a Brownian motion with a drift we proved the behavior of TV c on intervals going to infinity. Again, we obtained a LLN and CLT, though in this case they have a different interpretation and were easier to prove.All the results above were obtained in a functional setting, viz. we worked with processes describing the growth of the truncated variation in time. Moreover, in the same respect we also treated two closely related quantities -the so-called upward truncated variation and downward truncated variation.

Using Vovk's outer measure, which corresponds to a minimal superhedging price, the existence of quadratic variation is shown for "typical price paths" in the space of càdlàg functions possessing a mild restriction on the jumps directed downwards. In particular, this result includes the existence of quadratic variation of "typical price paths" in the space of non-negative càdlàg paths and implies the existence of quadratic variation in the sense of Föllmer quasi surely under all martingale measures. Based on the robust existence of the quadratic variation, a model-free Itô integration is developed.

Summary. We introduce the concept of truncated variation of Brownian motion with drift, which differs from regular variation by neglecting small jumps (smaller than some c > 0). We estimate the expected value of the truncated variation. The behaviour resembling phase transition as c varies is revealed. Truncated variation appears in the formula for an upper bound for return from any trading based on a single asset with flat commission.1. Introduction. Let (W t , t ≥ 0) be a Wiener process on the interval [0, T ] with drift µ, W t = µt + B t , where (B t , t ≥ 0) is a standard Brownian motion.It is well known (cf. [4]) that for any a < b, the variation of this process on [a, b] is infinite: