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

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

Medial patellofemoral ligament injury patterns and associated pathology in lateral patella dislocation: an MRI study

BACKGROUND: Lateral Patella dislocations are common injuries seen in the active and young adult populations. Our study focus was to evaluate medial patellofemoral ligament (MPFL) injury patterns and associated knee pathology using Magnetic Resonance Imaging studies. METHODS: MRI studies taken at one imaging site between January, 2007 to January, 2008 with the final diagnosis of patella dislocation were screened for this study. Of the 324 cases that were found, 195 patients with lateral patellar dislocation traumatic enough to cause bone bruises on the lateral femoral trochlea and the medial facet of the patella were selected for this study. The MRI images were reviewed by three independent observers for location and type of MPFL injury, osteochondral defects, loose bodies, MCL and meniscus tears. The data was analyzed as a single cohort and by gender. RESULTS: This study consisted of 127 males and 68 females; mean age of 23 yrs. Tear of the MPFL at the patellar attachment occurred in 93/195 knees (47%), at the femoral attachment in 50/195 knees (26%), and at both the femoral and patella attachment sites in 26/195 knees (13%). Attenuation of the MPFL without rupture occurred in 26/195 knees (13%). Associated findings included loose bodies in 23/195 (13%), meniscus tears 41/195 (21%), patella avulsion/fracture in 14/195 (7%), medial collateral ligament sprains/tears in 37/195 (19%) and osteochondral lesions in 96/195 knees (49%). Statistical analysis showed females had significantly more associated meniscus tears than the males (27% vs. 17%, p = 0.04). Although not statistically significant, osteochondral lesions were seen more in male patients with acute patella dislocation (52% vs. 42%, p = 0.08). CONCLUSION: Patients who present with lateral patella dislocation with the classic bone bruise pattern seen on MRI will likely rupture the MPFL at the patellar side. Females are more likely to have an associated meniscal tear than males; however, more males have underlying osteochondral lesions. Given the high percentage of associated pathology, we recommend a MRI of the knee in all patients who present with acute patella dislocation

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

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

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

Origin and insertion of the medial patellofemoral ligament: a systematic review of anatomy.

PURPOSE: The medial patellofemoral ligament (MPFL) is the major medial soft-tissue stabiliser of the patella, originating from the medial femoral condyle and inserting onto the medial patella. The exact position reported in the literature varies. Understanding the true anatomical origin and insertion of the MPFL is critical to successful reconstruction. The purpose of this systematic review was to determine these locations. METHODS: A systematic search of published (AMED, CINAHL, MEDLINE, EMBASE, PubMed and Cochrane Library) and unpublished literature databases was conducted from their inception to the 3 February 2016. All papers investigating the anatomy of the MPFL were eligible. Methodological quality was assessed using a modified CASP tool. A narrative analysis approach was adopted to synthesise the findings. RESULTS: After screening and review of 2045 papers, a total of 67 studies investigating the relevant anatomy were included. From this, the origin appears to be from an area rather than (as previously reported) a single point on the medial femoral condyle. The weighted average length was 56 mm with an 'hourglass' shape, fanning out at both ligament ends. CONCLUSION: The MPFL is an hourglass-shaped structure running from a triangular space between the adductor tubercle, medial femoral epicondyle and gastrocnemius tubercle and inserts onto the superomedial aspect of the patella. Awareness of anatomy is critical for assessment, anatomical repair and successful surgical patellar stabilisation. LEVEL OF EVIDENCE: Systematic review of anatomical dissections and imaging studies, Level IV