B-Spline Based Methods: From Monotone Multigrid Schemes for American Options to Uncertain Volatility Models

In the first part of this thesis, we consider B-spline based methods for pricing American options in the Black-Scholes and Heston model. The difference between these two models is the assumption on the volatility of the underlying asset. While in the Black-Scholes model the volatility is assumed to be constant, the Heston model includes a stochastic volatility variable. The underlying problems are formulated as parabolic variational inequalities. Recall that, in finance, to determine optimal risk strategies, one is not only interested in the solution of the variational inequality, i.e., the option price, but also in its partial derivatives up to order two, the so-called Greeks. A special feature for these option price problems is that initial conditions are typically given as piecewise linear continuous functions. Consequently, we have derived a spatial discretization based on cubic B-splines with coinciding knots at the points where the initial condition is not differentiable. Together with an implicit time stepping scheme, this enables us to achieve an accurate pointwise approximation of the partial derivatives up to order two. For the efficient numerical solution of the discrete variational inequality, we propose a monotone multigrid method for (tensor product) B-splines with possible internal coinciding knots. Corresponding numerical results show that the monotone multigrid method is robust with respect to the refinement level and mesh size. In the second part of this thesis, we consider the pricing of a European option in the uncertain volatility model. In this model the volatility of the underlying asset is a priori unknown and is assumed to lie within a range of extreme values. Mathematically, this problem can be formulated as a one dimensional parabolic Hamilton-Jacobi-Bellman equation and is also called Black-Scholes-Barenblatt equation. In the resulting non-linear equation, the diffusion coefficient is given by a volatility function which depends pointwise on the second derivative. This kind of non-linear partial differential equation does not admit a weak H^1-formulation. This is due to the fact that the non-linearity depends pointwise on the second derivative of the solution and, thus, no integration by parts is possible to pass the partial derivative onto a test function. But in the discrete setting this pointwise second derivative can be approximated in H^1 by L^1-normalized B-splines. It turns out that the approximation of the volatility function leads to discontinuities in the partial derivatives. In order to improve the approximation of the solution and its partial derivatives for cubic B-splines, we develop a Newton like algorithm within a knot insertion step. Corresponding numerical results show that the convergence of the solution and its partial derivatives are nearly optimal in the L^2-norm, when the location of volatility change is approximated with desired accuracy

Effect of Hydrocortisone on Mortality and Organ Support in Patients With Severe COVID-19: The REMAP-CAP COVID-19 Corticosteroid Domain Randomized Clinical Trial.

Importance: Evidence regarding corticosteroid use for severe coronavirus disease 2019 (COVID-19) is limited. Objective: To determine whether hydrocortisone improves outcome for patients with severe COVID-19. Design, Setting, and Participants: An ongoing adaptive platform trial testing multiple interventions within multiple therapeutic domains, for example, antiviral agents, corticosteroids, or immunoglobulin. Between March 9 and June 17, 2020, 614 adult patients with suspected or confirmed COVID-19 were enrolled and randomized within at least 1 domain following admission to an intensive care unit (ICU) for respiratory or cardiovascular organ support at 121 sites in 8 countries. Of these, 403 were randomized to open-label interventions within the corticosteroid domain. The domain was halted after results from another trial were released. Follow-up ended August 12, 2020. Interventions: The corticosteroid domain randomized participants to a fixed 7-day course of intravenous hydrocortisone (50 mg or 100 mg every 6 hours) (n = 143), a shock-dependent course (50 mg every 6 hours when shock was clinically evident) (n = 152), or no hydrocortisone (n = 108). Main Outcomes and Measures: The primary end point was organ support-free days (days alive and free of ICU-based respiratory or cardiovascular support) within 21 days, where patients who died were assigned -1 day. The primary analysis was a bayesian cumulative logistic model that included all patients enrolled with severe COVID-19, adjusting for age, sex, site, region, time, assignment to interventions within other domains, and domain and intervention eligibility. Superiority was defined as the posterior probability of an odds ratio greater than 1 (threshold for trial conclusion of superiority >99%). Results: After excluding 19 participants who withdrew consent, there were 384 patients (mean age, 60 years; 29% female) randomized to the fixed-dose (n = 137), shock-dependent (n = 146), and no (n = 101) hydrocortisone groups; 379 (99%) completed the study and were included in the analysis. The mean age for the 3 groups ranged between 59.5 and 60.4 years; most patients were male (range, 70.6%-71.5%); mean body mass index ranged between 29.7 and 30.9; and patients receiving mechanical ventilation ranged between 50.0% and 63.5%. For the fixed-dose, shock-dependent, and no hydrocortisone groups, respectively, the median organ support-free days were 0 (IQR, -1 to 15), 0 (IQR, -1 to 13), and 0 (-1 to 11) days (composed of 30%, 26%, and 33% mortality rates and 11.5, 9.5, and 6 median organ support-free days among survivors). The median adjusted odds ratio and bayesian probability of superiority were 1.43 (95% credible interval, 0.91-2.27) and 93% for fixed-dose hydrocortisone, respectively, and were 1.22 (95% credible interval, 0.76-1.94) and 80% for shock-dependent hydrocortisone compared with no hydrocortisone. Serious adverse events were reported in 4 (3%), 5 (3%), and 1 (1%) patients in the fixed-dose, shock-dependent, and no hydrocortisone groups, respectively. Conclusions and Relevance: Among patients with severe COVID-19, treatment with a 7-day fixed-dose course of hydrocortisone or shock-dependent dosing of hydrocortisone, compared with no hydrocortisone, resulted in 93% and 80% probabilities of superiority with regard to the odds of improvement in organ support-free days within 21 days. However, the trial was stopped early and no treatment strategy met prespecified criteria for statistical superiority, precluding definitive conclusions. Trial Registration: ClinicalTrials.gov Identifier: NCT02735707

Effect of angiotensin-converting enzyme inhibitor and angiotensin receptor blocker initiation on organ support-free days in patients hospitalized with COVID-19

IMPORTANCE Overactivation of the renin-angiotensin system (RAS) may contribute to poor clinical outcomes in patients with COVID-19. Objective To determine whether angiotensin-converting enzyme (ACE) inhibitor or angiotensin receptor blocker (ARB) initiation improves outcomes in patients hospitalized for COVID-19. DESIGN, SETTING, AND PARTICIPANTS In an ongoing, adaptive platform randomized clinical trial, 721 critically ill and 58 non–critically ill hospitalized adults were randomized to receive an RAS inhibitor or control between March 16, 2021, and February 25, 2022, at 69 sites in 7 countries (final follow-up on June 1, 2022). INTERVENTIONS Patients were randomized to receive open-label initiation of an ACE inhibitor (n = 257), ARB (n = 248), ARB in combination with DMX-200 (a chemokine receptor-2 inhibitor; n = 10), or no RAS inhibitor (control; n = 264) for up to 10 days. MAIN OUTCOMES AND MEASURES The primary outcome was organ support–free days, a composite of hospital survival and days alive without cardiovascular or respiratory organ support through 21 days. The primary analysis was a bayesian cumulative logistic model. Odds ratios (ORs) greater than 1 represent improved outcomes. RESULTS On February 25, 2022, enrollment was discontinued due to safety concerns. Among 679 critically ill patients with available primary outcome data, the median age was 56 years and 239 participants (35.2%) were women. Median (IQR) organ support–free days among critically ill patients was 10 (–1 to 16) in the ACE inhibitor group (n = 231), 8 (–1 to 17) in the ARB group (n = 217), and 12 (0 to 17) in the control group (n = 231) (median adjusted odds ratios of 0.77 [95% bayesian credible interval, 0.58-1.06] for improvement for ACE inhibitor and 0.76 [95% credible interval, 0.56-1.05] for ARB compared with control). The posterior probabilities that ACE inhibitors and ARBs worsened organ support–free days compared with control were 94.9% and 95.4%, respectively. Hospital survival occurred in 166 of 231 critically ill participants (71.9%) in the ACE inhibitor group, 152 of 217 (70.0%) in the ARB group, and 182 of 231 (78.8%) in the control group (posterior probabilities that ACE inhibitor and ARB worsened hospital survival compared with control were 95.3% and 98.1%, respectively). CONCLUSIONS AND RELEVANCE In this trial, among critically ill adults with COVID-19, initiation of an ACE inhibitor or ARB did not improve, and likely worsened, clinical outcomes. TRIAL REGISTRATION ClinicalTrials.gov Identifier: NCT0273570

Analysis of Potential Biomarkers and Modifier Genes Affecting the Clinical Course of CLN3 Disease

Mutations in the CLN3 gene lead to juvenile neuronal ceroid lipofuscinosis, a pediatric neurodegenerative disorder characterized by visual loss, epilepsy and psychomotor deterioration. Although most CLN3 patients carry the same 1-kb deletion in the CLN3 gene, their disease phenotype can be variable. The aims of this study were to (i) study the clinical phenotype in CLN3 patients with identical genotype, (ii) identify genes that are dysregulated in CLN3 disease regardless of the clinical course that could be useful as biomarkers, and (iii) find modifier genes that affect the progression rate of the disease. A total of 25 CLN3 patients homozygous for the 1-kb deletion were classified into groups with rapid, average or slow disease progression using an established clinical scoring system. Genome-wide expression profiling was performed in eight CLN3 patients with different disease progression and matched controls. The study showed high phenotype variability in CLN3 patients. Five genes were dysregulated in all CLN3 patients and present candidate biomarkers of the disease. Of those, dual specificity phosphatase 2 (DUSP2) was also validated in acutely CLN3-depleted cell models and in CbCln3Δex7/8 cerebellar precursor cells. A total of 13 genes were upregulated in patients with rapid disease progression and downregulated in patients with slow disease progression; one gene showed dysregulation in the opposite way. Among these potential modifier genes, guanine nucleotide exchange factor 1 for small GTPases of the Ras family (RAPGEF1) and transcription factor Spi-B (SPIB) were validated in an acutely CLN3-depleted cell model. These findings indicate that differential perturbations of distinct signaling pathways might alter disease progression and provide insight into the molecular alterations underlying neuronal dysfunction in CLN3 disease and neurodegeneration in general

A Bayesian reanalysis of the Standard versus Accelerated Initiation of Renal-Replacement Therapy in Acute Kidney Injury (STARRT-AKI) trial

Background Timing of initiation of kidney-replacement therapy (KRT) in critically ill patients remains controversial. The Standard versus Accelerated Initiation of Renal-Replacement Therapy in Acute Kidney Injury (STARRT-AKI) trial compared two strategies of KRT initiation (accelerated versus standard) in critically ill patients with acute kidney injury and found neutral results for 90-day all-cause mortality. Probabilistic exploration of the trial endpoints may enable greater understanding of the trial findings. We aimed to perform a reanalysis using a Bayesian framework. Methods We performed a secondary analysis of all 2927 patients randomized in multi-national STARRT-AKI trial, performed at 168 centers in 15 countries. The primary endpoint, 90-day all-cause mortality, was evaluated using hierarchical Bayesian logistic regression. A spectrum of priors includes optimistic, neutral, and pessimistic priors, along with priors informed from earlier clinical trials. Secondary endpoints (KRT-free days and hospital-free days) were assessed using zero–one inflated beta regression. Results The posterior probability of benefit comparing an accelerated versus a standard KRT initiation strategy for the primary endpoint suggested no important difference, regardless of the prior used (absolute difference of 0.13% [95% credible interval [CrI] − 3.30%; 3.40%], − 0.39% [95% CrI − 3.46%; 3.00%], and 0.64% [95% CrI − 2.53%; 3.88%] for neutral, optimistic, and pessimistic priors, respectively). There was a very low probability that the effect size was equal or larger than a consensus-defined minimal clinically important difference. Patients allocated to the accelerated strategy had a lower number of KRT-free days (median absolute difference of − 3.55 days [95% CrI − 6.38; − 0.48]), with a probability that the accelerated strategy was associated with more KRT-free days of 0.008. Hospital-free days were similar between strategies, with the accelerated strategy having a median absolute difference of 0.48 more hospital-free days (95% CrI − 1.87; 2.72) compared with the standard strategy and the probability that the accelerated strategy had more hospital-free days was 0.66. Conclusions In a Bayesian reanalysis of the STARRT-AKI trial, we found very low probability that an accelerated strategy has clinically important benefits compared with the standard strategy. Patients receiving the accelerated strategy probably have fewer days alive and KRT-free. These findings do not support the adoption of an accelerated strategy of KRT initiation

Effect of Antiplatelet Therapy on Survival and Organ Support–Free Days in Critically Ill Patients With COVID-19

International audienc