Fractional smoothness and applications in finance

This overview article concerns the notion of fractional smoothness of random variables of the form $g(X_T)$, where $X=(X_t)_{t\in [0,T]}$ is a certain diffusion process. We review the connection to the real interpolation theory, give examples and applications of this concept. The applications in stochastic finance mainly concern the analysis of discrete time hedging errors. We close the review by indicating some further developments.Comment: Chapter of AMAMEF book. 20 pages

The Euler-Maruyama approximation for the absorption time of the CEV diffusion

A standard convergence analysis of the simulation schemes for the hitting times of diffusions typically requires non-degeneracy of their coefficients on the boundary, which excludes the possibility of absorption. In this paper we consider the CEV diffusion from the mathematical finance and show how a weakly consistent approximation for the absorption time can be constructed, using the Euler-Maruyama scheme

Construction of a Mean Square Error Adaptive Euler--Maruyama Method with Applications in Multilevel Monte Carlo

A formal mean square error expansion (MSE) is derived for Euler--Maruyama numerical solutions of stochastic differential equations (SDE). The error expansion is used to construct a pathwise a posteriori adaptive time stepping Euler--Maruyama method for numerical solutions of SDE, and the resulting method is incorporated into a multilevel Monte Carlo (MLMC) method for weak approximations of SDE. This gives an efficient MSE adaptive MLMC method for handling a number of low-regularity approximation problems. In low-regularity numerical example problems, the developed adaptive MLMC method is shown to outperform the uniform time stepping MLMC method by orders of magnitude, producing output whose error with high probability is bounded by TOL>0 at the near-optimal MLMC cost rate O(TOL^{-2}log(TOL)^4).Comment: 43 pages, 12 figure

Multilevel Monte Carlo methods

The author's presentation of multilevel Monte Carlo path simulation at the MCQMC 2006 conference stimulated a lot of research into multilevel Monte Carlo methods. This paper reviews the progress since then, emphasising the simplicity, flexibility and generality of the multilevel Monte Carlo approach. It also offers a few original ideas and suggests areas for future research

Magnetic resonance imaging of the erector spinae muscles in Duchenne muscular dystrophy: implication for scoliotic deformities

<p>Abstract</p> <p>Background</p> <p>In Duchenne muscular dystrophy (DMD), the muscular degeneration often leads to the development of scoliosis. Our objective was to investigate how anatomical changes in back muscles can lead to scoliosis. Muscular volume and the level of fat infiltration in those muscles were thus evaluated, in non-scoliotic, pre-scoliotic and scoliotic patients. The overlying skin thickness over the apex level of scoliotic deformations was also measured to facilitate the interpretation of electromyographic signals when recorded on the skin surface.</p> <p>Methods</p> <p>In 8 DMD patients and two healthy controls with no known muscular deficiencies, magnetic resonance imaging (MRI) was used to measure continuously at 3 mm intervals the distribution of the erector spinae (ES) muscle in the T8-L4 region as well as fat infiltration in the muscle and overlying skin thickness: four patients were non-scoliotic (NS), two were pre-scoliotic (PS, Cobb angle < 15°) and two were scoliotic (S, Cobb angle ≥ 15°). For each subject, 63 images 3 mm thick of the ES muscle were obtained in the T8-L4 region on both sides of the spine. The pixel dimension was 0.39 × 0.39 mm. With a commercial software, on each 12 bits image, the ES contour on the left and on the right sides of the spine were manually determined as well as those of its constituents i.e., the iliocostalis (IL), the longissimus (LO) and the spinalis (SP) muscles. Following this segmentation, the surfaces within the contours were determined, the muscles volume were obtained, the amount of fat infiltration inside each muscle was evaluated and the overlying skin thickness measured.</p> <p>Findings</p> <p>The volume of the ES muscle of our S and PS patients was found smaller on the convex side relative to the concave one by 5.3 ± 0.7% and 2.8 ± 0.2% respectively. For the 4 NS patients, the volume difference of this muscle between right and left sides was 2.1 ± 1.5% and for the 2 controls, it was 1.4 ± 1.2%. Fat infiltration for the S and the PS patients was larger on the convex side than on the concave one (4.4 ± 1.6% and 4.5 ± 0.7% respectively) and the difference was more important near the apex. Infiltration was more important in the lateral IL muscle than in the medial SP and it was always larger near L2 than at any other spinal level. Fat infiltration was much more important in the ES for the DMD patients (49.9% ± 1.6%) than for the two controls (2.6 ± 0.8%). As for the overlying skin thickness measured near the deformity of the patients, it was larger on the concave than on the convex side: 14.8 ± 6.1 vs 13.5 ± 5.7 mm for the S and 10.3 ± 6.3 vs 9.8 ± 5.6 mm for the PS.</p> <p>Interpretation</p> <p>In DMD patients, our results indicate that a larger replacement of muscles fibers by fat infiltration on one side of the spine is a factor that can lead to the development of scoliosis. Efforts to slow such an infiltration on the most affected side of the spine could thus be beneficial to those patients by delaying the apparition of the scoliotic deformation. In addition to anatomical considerations, results obtained from the same patients but in experiments dealing with electromyography recordings, point to differences in the muscular contraction mechanisms and/or of the neural input to back muscles. This is similar to the adolescent idiopathic scoliosis (AIS) where a role of the nervous system in the development of the deformation has also been suggested.</p

Atypical Social Modulation of Imitation in Autism Spectrum Conditions.

Appropriate modulation of imitation according to social context is important for successful social interaction. In the present study we subliminally primed high-functioning adults with ASC and age- and IQ-matched controls with either a pro- or non- social attitude. Following priming, an automatic imitation paradigm was used to acquire an index of imitation. Whereas imitation levels were higher for pro-socially primed relative to non-socially primed control participants, there was no difference between pro- and non- socially primed individuals with ASC. We conclude that high-functioning adults with ASC demonstrate atypical social modulation of imitation. Given the importance of imitation in social interaction we speculate that difficulties with the modulation of imitation may contribute to the social problems characteristic of ASC

Contorted and ordinary body postures in the human brain

Social interaction and comprehension of non-verbal behaviour requires a representation of people’s bodies. Research into the neural underpinnings of body representation implicates several brain regions including extrastriate and fusiform body areas (EBA and FBA), superior temporal sulcus (STS), inferior frontal gyrus (IFG) and inferior parietal lobule (IPL). The different roles played by these regions in parsing familiar and unfamiliar body postures remain unclear. We examined the responses of this body observation network to static images of ordinary and contorted postures by using a repetition suppression design in functional neuroimaging. Participants were scanned whilst observing static images of a contortionist or a group of objects in either ordinary or unusual configurations, presented from different viewpoints. Greater activity emerged in EBA and FBA when participants viewed contorted compared to ordinary body postures. Repeated presentation of the same posture from different viewpoints lead to suppressed responses in the fusiform gyrus as well as three regions that are characteristically activated by observing moving bodies, namely STS, IFG and IPL. These four regions did not distinguish the image viewpoint or the plausibility of the posture. Together, these data define a broad cortical network for processing static body postures, including regions classically associated with action observation

Multilevel Monte Carlo for exponential Lévy models

We apply the multilevel Monte Carlo method for option pricing problems using exponential Lévy models with a uniform timestep discretisation. For lookback and barrier options, we derive estimates of the convergence rate of the error introduced by the discrete monitoring of the running supremum of a broad class of Lévy processes. We then use these to obtain upper bounds on the multilevel Monte Carlo variance convergence rate for the Variance Gamma, NIG and a-stable processes. We also provide analysis of a trapezoidal approximation for Asian options. Our method is illustrated by numerical experiments

Assessment of the paraspinal muscles of subjects presenting an idiopathic scoliosis: an EMG pilot study

BACKGROUND: It is known that the back muscles of scoliotic subjects present abnormalities in their fiber type composition. Some researchers have hypothesized that abnormal fiber composition can lead to paraspinal muscle dysfunction such as poor neuromuscular efficiency and muscle fatigue. EMG parameters were used to evaluate these impairments. The purpose of the present study was to examine the clinical potential of different EMG parameters such as amplitude (RMS) and median frequency (MF) of the power spectrum in order to assess the back muscles of patients presenting idiopathic scoliosis in terms of their neuromuscular efficiency and their muscular fatigue. METHODS: L5/S1 moments during isometric efforts in extension were measured in six subjects with idiopathic scoliosis and ten healthy controls. The subjects performed three 7 s ramp contractions ranging from 0 to 100% maximum voluntary contraction (MVC) and one 30 s sustained contraction at 75% MVC. Surface EMG activity was recorded bilaterally from the paraspinal muscles at L5, L3, L1 and T10. The slope of the EMG RMS/force (neuromuscular efficiency) and MF/force (muscle composition) relationships were computed during the ramp contractions while the slope of the EMG RMS/time and MF/time relationships (muscle fatigue) were computed during the sustained contraction. Comparisons were performed between the two groups and between the left and right sides for the EMG parameters. RESULTS: No significant group or side differences between the slopes of the different measures used were found at the level of the apex (around T10) of the major curve of the spine. However, a significant side difference was seen at a lower level (L3, p = 0.01) for the MF/time parameter. CONCLUSION: The EMG parameters used in this study could not discriminate between the back muscles of scoliotic subjects and those of control subject regarding fiber type composition, neuromuscular efficiency and muscle fatigue at the level of the apex. The results of this pilot study indicate that compensatory strategies are potentially seen at lower level of the spine with these EMG parameters