A Differentiation Theory for It\^o's Calculus

A peculiar feature of It\^o's calculus is that it is an integral calculus that gives no explicit derivative with a systematic differentiation theory counterpart, as in elementary calculus. So, can we define a pathwise stochastic derivative of semimartingales with respect to Brownian motion that leads to a differentiation theory counterpart to It\^o's integral calculus? From It\^o's definition of his integral, such a derivative must be based on the quadratic covariation process. We give such a derivative in this note and we show that it leads to a fundamental theorem of stochastic calculus, a generalized stochastic chain rule that includes the case of convex functions acting on continuous semimartingales, and the stochastic mean value and Rolle's theorems. In addition, it interacts with basic algebraic operations on semimartingales similarly to the way the deterministic derivative does on deterministic functions, making it natural for computations. Such a differentiation theory leads to many interesting applications some of which we address in an upcoming article.Comment: 10 pages, 9/9 papers from my 2000-2006 collection. I proved these results and more earlier in 2004. I generalize this theory in upcoming articles. I also apply this theory in upcoming article

A note on a result of Liptser-Shiryaev

Given two stochastic equations with different drift terms, under very weak assumptions Liptser and Shiryaev provide the equivalence of the laws of the solutions to these equations by means of Girsanov transform. Their assumptions involve both the drift terms. We are interested in the same result but with the main assumption involving only the difference of the drift terms. Applications of our result will be presented in the finite as well as in the infinite dimensional setting.Comment: 22 pages; revised and enlarged versio

Composition of processes and related partial differential equations

In this paper different types of compositions involving independent fractional Brownian motions B^j_{H_j}(t), t>0, j=1,$ are examined. The partial differential equations governing the distributions of I_F(t)=B^1_{H_1}(|B^2_{H_2}(t)|), t>0 and J_F(t)=B^1_{H_1}(|B^2_{H_2}(t)|^{1/H_1}), t>0 are derived by different methods and compared with those existing in the literature and with those related to B^1(|B^2_{H_2}(t)|), t>0. The process of iterated Brownian motion I^n_F(t), t>0 is examined in detail and its moments are calculated. Furthermore for J^{n-1}_F(t)=B^1_{H}(|B^2_H(...|B^n_H(t)|^{1/H}...)|^{1/H}), t>0 the following factorization is proved J^{n-1}_F(t)=\prod_{j=1}^{n} B^j_{\frac{H}{n}}(t), t>0. A series of compositions involving Cauchy processes and fractional Brownian motions are also studied and the corresponding non-homogeneous wave equations are derived.Comment: 32 page

Disease-specific variant pathogenicity prediction significantly improves variant interpretation in inherited cardiac conditions

Funder: Science and Technology Development Fund; doi: https://doi.org/10.13039/Funder: Al-Alfi FoundationFunder: Magdi Yacoub Heart FoundationFunder: Rosetrees and Stoneygate Imperial College Research FellowshipFunder: National Health and Medical Research Council (Australia)Abstract: Purpose: Accurate discrimination of benign and pathogenic rare variation remains a priority for clinical genome interpretation. State-of-the-art machine learning variant prioritization tools are imprecise and ignore important parameters defining gene–disease relationships, e.g., distinct consequences of gain-of-function versus loss-of-function variants. We hypothesized that incorporating disease-specific information would improve tool performance. Methods: We developed a disease-specific variant classifier, CardioBoost, that estimates the probability of pathogenicity for rare missense variants in inherited cardiomyopathies and arrhythmias. We assessed CardioBoost’s ability to discriminate known pathogenic from benign variants, prioritize disease-associated variants, and stratify patient outcomes. Results: CardioBoost has high global discrimination accuracy (precision recall area under the curve [AUC] 0.91 for cardiomyopathies; 0.96 for arrhythmias), outperforming existing tools (4–24% improvement). CardioBoost obtains excellent accuracy (cardiomyopathies 90.2%; arrhythmias 91.9%) for variants classified with >90% confidence, and increases the proportion of variants classified with high confidence more than twofold compared with existing tools. Variants classified as disease-causing are associated with both disease status and clinical severity, including a 21% increased risk (95% confidence interval [CI] 11–29%) of severe adverse outcomes by age 60 in patients with hypertrophic cardiomyopathy. Conclusions: A disease-specific variant classifier outperforms state-of-the-art genome-wide tools for rare missense variants in inherited cardiac conditions (https://www.cardiodb.org/cardioboost/), highlighting broad opportunities for improved pathogenicity prediction through disease specificity

Disease-specific variant pathogenicity prediction significantly improves variant interpretation in inherited cardiac conditions

Funder: Science and Technology Development Fund; doi: https://doi.org/10.13039/Funder: Al-Alfi FoundationFunder: Magdi Yacoub Heart FoundationFunder: Rosetrees and Stoneygate Imperial College Research FellowshipFunder: National Health and Medical Research Council (Australia)Abstract: Purpose: Accurate discrimination of benign and pathogenic rare variation remains a priority for clinical genome interpretation. State-of-the-art machine learning variant prioritization tools are imprecise and ignore important parameters defining gene–disease relationships, e.g., distinct consequences of gain-of-function versus loss-of-function variants. We hypothesized that incorporating disease-specific information would improve tool performance. Methods: We developed a disease-specific variant classifier, CardioBoost, that estimates the probability of pathogenicity for rare missense variants in inherited cardiomyopathies and arrhythmias. We assessed CardioBoost’s ability to discriminate known pathogenic from benign variants, prioritize disease-associated variants, and stratify patient outcomes. Results: CardioBoost has high global discrimination accuracy (precision recall area under the curve [AUC] 0.91 for cardiomyopathies; 0.96 for arrhythmias), outperforming existing tools (4–24% improvement). CardioBoost obtains excellent accuracy (cardiomyopathies 90.2%; arrhythmias 91.9%) for variants classified with >90% confidence, and increases the proportion of variants classified with high confidence more than twofold compared with existing tools. Variants classified as disease-causing are associated with both disease status and clinical severity, including a 21% increased risk (95% confidence interval [CI] 11–29%) of severe adverse outcomes by age 60 in patients with hypertrophic cardiomyopathy. Conclusions: A disease-specific variant classifier outperforms state-of-the-art genome-wide tools for rare missense variants in inherited cardiac conditions (https://www.cardiodb.org/cardioboost/), highlighting broad opportunities for improved pathogenicity prediction through disease specificity