Numerical Solutions of Backward Stochastic Differential Equations: A Finite Transposition Method

In this note, we present a new numerical method for solving backward stochastic differential equations. Our method can be viewed as an analogue of the classical finite element method solving deterministic partial differential equations.Comment: 4 page

A note on "Problem of eigenvalues of stochastic Hamiltonian systems with boundary conditions"

The eigenvalue problem of stochastic Hamiltonian systems with boundary conditions was studied by Peng \cite{peng} in 2000. For one-dimensional case, denoting by $\{\lambda_n\}_{n=1}^{\infty}$ all the eigenvalues of such an eigenvalue problem, Peng proved that $\lambda_n\to +\infty$. In this short note, we prove that the growth order of $\lambda_n$ is the same as $n^2$ as $n\to +\infty$. Apart from the interesting of its own, by this result, the statistic period of solutions of FBSDEs can be estimated directly by corresponding coefficients and time duration.Comment: 7 page

Essentially normal quotient weighted Bergman modules over the bidisk and distinguished varieties

We introduce a Grassmannian structure for a class of quotient Hilbert modules and attack the polydisc version of Arveson-Douglas conjecture associated to distinguished varieties. More interestingly, we obtain an operator-theoretic characterization of distinguished varieties in the bidisk in terms of essential normality of the quotient modules. As an application, we study the K-homology of the boundary of distinguished variety

Benchmarking Software Vulnerability Detection Techniques: A Survey

Software vulnerabilities can have serious consequences, which is why many techniques have been proposed to defend against them. Among these, vulnerability detection techniques are a major area of focus. However, there is a lack of a comprehensive approach for benchmarking these proposed techniques. In this paper, we present the first survey that comprehensively investigates and summarizes the current state of software vulnerability detection benchmarking. We review the current literature on benchmarking vulnerability detection, including benchmarking approaches in technique-proposing papers and empirical studies. We also separately discuss the benchmarking approaches for traditional and deep learning-based vulnerability detection techniques. Our survey analyzes the challenges of benchmarking software vulnerability detection techniques and the difficulties involved. We summarize the challenges of benchmarking software vulnerability detection techniques and describe possible solutions for addressing these challenges