Flows and stochastic Taylor series in Ito calculus

For stochastic systems driven by continuous semimartingales an explicit formula for the logarithm of the Ito flow map is given. A similar formula is also obtained for solutions of linear matrix-valued SDEs driven by arbitrary semimartingales. The computation relies on the lift to quasi-shuffle algebras of formulas involving products of Ito integrals of semimartingales. Whereas the Chen-Strichartz formula computing the logarithm of the Stratonovich flow map is classically expanded as a formal sum indexed by permutations, the analogous formula in Ito calculus is naturally indexed by surjections. This reflects the change of algebraic background involved in the transition between the two integration theories

Levy Processes and Quasi-Shuffle Algebras

We investigate the algebra of repeated integrals of semimartingales. We prove that a minimal family of semimartingales generates a quasi-shuffle algebra. In essence, to fulfill the minimality criterion, first, the family must be a minimal generator of the algebra of repeated integrals generated by its elements and by quadratic covariation processes recursively constructed from the elements of the family. Second, recursively constructed quadratic covariation processes may lie in the linear span of previously constructed ones and of the family, but may not lie in the linear span of repeated integrals of these. We prove that a finite family of independent Levy processes that have finite moments generates a minimal family. Key to the proof are the Teugels martingales and a strong orthogonalization of them. We conclude that a finite family of independent Levy processes form a quasi-shuffle algebra. We discuss important potential applications to constructing efficient numerical methods for the strong approximation of stochastic differential equations driven by Levy processes.Comment: 10 page

Algebraic Structures and Stochastic Differential Equations driven by Levy processes

We construct an efficient integrator for stochastic differential systems driven by Levy processes. An efficient integrator is a strong approximation that is more accurate than the corresponding stochastic Taylor approximation, to all orders and independent of the governing vector fields. This holds provided the driving processes possess moments of all orders and the vector fields are sufficiently smooth. Moreover the efficient integrator in question is optimal within a broad class of perturbations for half-integer global root mean-square orders of convergence. We obtain these results using the quasi-shuffle algebra of multiple iterated integrals of independent Levy processes.Comment: 41 pages, 11 figure

Stochastic expansions and Hopf algebras

We study solutions to nonlinear stochastic differential systems driven by a multi-dimensional Wiener process. A useful algorithm for strongly simulating such stochastic systems is the Castell--Gaines method, which is based on the exponential Lie series. When the diffusion vector fields commute, it has been proved that at low orders this method is more accurate in the mean-square error than corresponding stochastic Taylor methods. However it has also been shown that when the diffusion vector fields do not commute, this is not true for strong order one methods. Here we prove that when there is no drift, and the diffusion vector fields do not commute, the exponential Lie series is usurped by the sinh-log series. In other words, the mean-square error associated with a numerical method based on the sinh-log series, is always smaller than the corresponding stochastic Taylor error, in fact to all orders. Our proof utilizes the underlying Hopf algebra structure of these series, and a two-alphabet associative algebra of shuffle and concatenation operations. We illustrate the benefits of the proposed series in numerical studies.Comment: 23 pages, 4 figure

Graph4Med: a web application and a graph database for visualizing and analyzing medical databases

Background: Medical databases normally contain large amounts of data in a variety of forms. Although they grant significant insights into diagnosis and treatment, implementing data exploration into current medical databases is challenging since these are often based on a relational schema and cannot be used to easily extract information for cohort analysis and visualization. As a consequence, valuable information regarding cohort distribution or patient similarity may be missed. With the rapid advancement of biomedical technologies, new forms of data from methods such as Next Generation Sequencing (NGS) or chromosome microarray (array CGH) are constantly being generated; hence it can be expected that the amount and complexity of medical data will rise and bring relational database systems to a limit. Description: We present Graph4Med, a web application that relies on a graph database obtained by transforming a relational database. Graph4Med provides a straightforward visualization and analysis of a selected patient cohort. Our use case is a database of pediatric Acute Lymphoblastic Leukemia (ALL). Along routine patients’ health records it also contains results of latest technologies such as NGS data. We developed a suitable graph data schema to convert the relational data into a graph data structure and store it in Neo4j. We used NeoDash to build a dashboard for querying and displaying patients’ cohort analysis. This way our tool (1) quickly displays the overview of patients’ cohort information such as distributions of gender, age, mutations (fusions), diagnosis; (2) provides mutation (fusion) based similarity search and display in a maneuverable graph; (3) generates an interactive graph of any selected patient and facilitates the identification of interesting patterns among patients. Conclusion: We demonstrate the feasibility and advantages of a graph database for storing and querying medical databases. Our dashboard allows a fast and interactive analysis and visualization of complex medical data. It is especially useful for patients similarity search based on mutations (fusions), of which vast amounts of data have been generated by NGS in recent years. It can discover relationships and patterns in patients cohorts that are normally hard to grasp. Expanding Graph4Med to more medical databases will bring novel insights into diagnostic and research

The Role of Social Isolation and the Development of Depression: A Comparison of the Widowed and Married Oldest Old in Germany

Widowhood is common in old age, can be accompanied by serious health consequences and is often linked to substantial changes in social network. Little is known about the impact of social isolation on the development of depressive symptoms over time taking widowhood into account. We provide results from the follow-up 5 to follow-up 9 from the longitudinal study AgeCoDe and its follow-up study AgeQualiDe. Depression was measured with GDS-15 and social isolation was assessed using the Lubben Social Network Scale (LSNS-6). The group was aligned of married and widowed people in old age and education through entropy balancing. Linear mixed models were used to examine the frequency of occurrence of depressive symptoms for widowed and married elderly people depending on the risk of social isolation. Our study shows that widowhood alone does not lead to an increased occurrence of depressive symptoms. However, "widowed oldest old", who are also at risk of social isolation, have significantly more depressive symptoms than those without risk. In the group of "married oldest old", women have significantly more depressive symptoms than men, but isolated and non-isolated do not differ. Especially for people who have lost a spouse, the social network changes significantly and increases the risk for social isolation. This represents a risk factor for the occurrence of depressive symptoms