A Probabilistic Scheme for Fully Nonlinear Nonlocal Parabolic PDEs with singular L\'evy measures

We introduce a Monte Carlo scheme for fully nonlinear parabolic nonlocal PDE's whose nonlinearity in of Hamilton-Jacobi-Bellman-Isaacs (HJBI for short). We avoid the difficulties of infinite L\'evy measure by truncation of the L\'evy integral. The first result provides the convergence of the scheme for general parabolic nonlinearities. The second result provides bounds on the rate of convergence for concave (or equivalently convex) nonlinearities. For both results, it is crucial to choose truncation of the infinite L\'evy measure appropriately dependent on the time discretization. We also introduce a Monte Carlo Quadrature method to approximate the nonlocal term in the HJBI nonlinearity.Comment: Keywords: Viscosity solution, nonlocal PDE, Monte Carlo approximatio

Asymptotic Analysis for Optimal Dividends in a Dual Risk Model

The dual risk model is a popular model in finance and insurance, which is often used to model the wealth process of a venture capital or high tech company. Optimal dividends have been extensively studied in the literature for the dual risk model. It is well known that the value function of this optimal control problem does not yield closed-form solutions except in some special cases. In this paper, we study the asymptotics of the optimal dividends problem when the parameters of the model go to either zero or infinity. Our results provide insights to the optimal strategies and the optimal values when the parameters are extreme.Comment: 23 page

Strong Convergence to the Homogenized Limit of Parabolic Equations with Random Coefficients

This paper is concerned with the study of solutions to discrete parabolic equations in divergence form with random coefficients, and their convergence to solutions of a homogenized equation. It has previously been shown that if the random environment is translational invariant and ergodic, then solutions of the random equation converge under diffusive scaling to solutions of a homogenized parabolic PDE. In this paper point-wise estimates are obtained on the difference between the averaged solution to the random equation and the solution to the homogenized equation for certain random environments which are strongly mixing.Comment: 46 page

A Probabilistic Numerical Method for Fully Nonlinear Parabolic PDEs

We consider the probabilistic numerical scheme for fully nonlinear PDEs suggested in \cite{cstv}, and show that it can be introduced naturally as a combination of Monte Carlo and finite differences scheme without appealing to the theory of backward stochastic differential equations. Our first main result provides the convergence of the discrete-time approximation and derives a bound on the discretization error in terms of the time step. An explicit implementable scheme requires to approximate the conditional expectation operators involved in the discretization. This induces a further Monte Carlo error. Our second main result is to prove the convergence of the latter approximation scheme, and to derive an upper bound on the approximation error. Numerical experiments are performed for the approximation of the solution of the mean curvature flow equation in dimensions two and three, and for two and five-dimensional (plus time) fully-nonlinear Hamilton-Jacobi-Bellman equations arising in the theory of portfolio optimization in financial mathematics

A stochastic approximation for fully nonlinear free boundary parabolic problems

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/106675/1/num21841.pd

Increasing Non-tuberculous Mycobacteria Infections in Veterans With COPD and Association With Increased Risk of Mortality

Background: There are limited data on the epidemiology of Non-tuberculous mycobacteria (NTM) infections among patients with COPD, particularly in the veteran population. This study examined the prevalence, incidence, and mortality of pulmonary NTM infections among veterans with COPD population in the United States.Methods: We analyzed nationwide data from Veterans Affairs Hospitals from 2001 to 2015. First, we determined the incidence and prevalence rates and geographic distribution of NTM infections among veterans with COPD and then we evaluated the association between NTM infections with mortality among veterans with COPD. Pulmonary NTM and COPD diagnosis were defined based on charting claims for each condition on ≥2 occasions and ≥30 days apart. COPD diagnoses had to precede diagnosis of NTM. Cox Proportional-Hazards Regression was performed to determine the dependency of survival time of COPD patients with NTM.Results: The incidence and prevalence rates of NTM rose over the study period, with a sharp rise in incidence after 2012. The areas with the highest NTM period prevalence were Puerto Rico (370), followed by Florida (351) and District of Columbia (309) in 100,000 COPD population. Mortality registered for those patients with COPD Patients and NTM infection was 1.43 times higher compared to those that were uninfected.Conclusions: NTM rates have been increasing in veterans with COPD since 2012. NTM infection is associated with increased risk of mortality. This highlights the importance of identifying preventable risk factors associated with NTM infections in subjects with COPD

Fuzz, Penetration, and AI Testing for SoC Security Verification: Challenges and Solutions

The ever-increasing usage and application of system-on-chips (SoCs) has resulted in the tremendous modernization of these architectures. For a modern SoC design, with the inclusion of numerous complex and heterogeneous intellectual properties (IPs), and its privacy-preserving declaration, there exists a wide variety of highly sensitive assets. These assets must be protected from any unauthorized access and against a diverse set of attacks. Attacks for obtaining such assets could be accomplished through different sources, including malicious IPs, malicious or vulnerable firmware/software, unreliable and insecure interconnection and communication protocol, and side-channel vulnerabilities through power/performance profiles. Any unauthorized access to such highly sensitive assets may result in either a breach of company secrets for original equipment manufactures (OEM) or identity theft for the end-user. Unlike the enormous advances in functional testing and verification of the SoC architecture, security verification is still on the rise, and little endeavor has been carried out by academia and industry. Unfortunately, there exists a huge gap between the modernization of the SoC architectures and their security verification approaches. With the lack of automated SoC security verification in modern electronic design automation (EDA) tools, we provide a comprehensive overview of the requirements that must be realized as the fundamentals of the SoC security verification process in this paper. By reviewing these requirements, including the creation of a unified language for SoC security verification, the definition of security policies, formulation of the security verification, etc., we put forward a realization of the utilization of self-refinement techniques, such as fuzz, penetration, and AI testing, for security verification purposes. We evaluate all the challenges and resolution possibilities, and we provide the potential approaches for the realization of SoC security verification via these self-refinement techniques