Optimality of Excess-Loss Reinsurance under a Mean-Variance Criterion

In this paper, we study an insurer's reinsurance-investment problem under a mean-variance criterion. We show that excess-loss is the unique equilibrium reinsurance strategy under a spectrally negative L\'{e}vy insurance model when the reinsurance premium is computed according to the expected value premium principle. Furthermore, we obtain the explicit equilibrium reinsurance-investment strategy by solving the extended Hamilton-Jacobi-Bellman equation

Localizing virtual structure sheaves by cosections

We construct a cosection localized virtual structure sheaf when a Deligne-Mumford stack is equipped with a perfect obstruction theory and a cosection of the obstruction sheaf.Comment: 25 pages; published versio

Partial Decode-Forward Relaying for the Gaussian Two-Hop Relay Network

The multicast capacity of the Gaussian two-hop relay network with one source, $N$ relays, and $L$ destinations is studied. It is shown that a careful modification of the partial decode--forward coding scheme, whereby the relays cooperate through degraded sets of message parts, achieves the cutset upper bound within $(1/2)\log N$ bits regardless of the channel gains and power constraints. This scheme improves upon a previous scheme by Chern and Ozgur, which is also based on partial decode--forward yet has an unbounded gap from the cutset bound for $L \ge 2$ destinations. When specialized to independent codes among relays, the proposed scheme achieves within $\log N$ bits from the cutset bound. The computation of this relaxation involves evaluating mutual information across $L(N+1)$ cuts out of the total $L 2^N$ possible cuts, providing a very simple linear-complexity algorithm to approximate the single-source multicast capacity of the Gaussian two-hop relay network.Comment: 7 pages (2 columns), 4 figures, submitted to the IEEE Transactions on Information Theor

Gromov-Witten invariants of varieties with holomorphic 2-forms

We show that a holomorphic two-form $\theta$ on a smooth algebraic variety X localizes the virtual fundamental class of the moduli of stable maps \mgn(X,\beta) to the locus where $\theta$ degenerates; it then enables us to define the localized GW-invariant, an algebro-geometric analogue of the local invariant of Lee and Parker in symplectic geometry, which coincides with the ordinary GW-invariant when X is proper. It is deformation invariant. Using this, we prove formulas for low degree GW-invariants of minimal general type surfaces with p_g>0 conjectured by Maulik and Pandharipande.Comment: 36 page

Categorification of Donaldson-Thomas invariants via Perverse Sheaves

We show that there is a perverse sheaf on a fine moduli space of stable sheaves on a smooth projective Calabi-Yau 3-fold, which is locally the perverse sheaf of vanishing cycles for a local Chern-Simons functional, possibly after taking an etale Galois cover. This perverse sheaf lifts to a mixed Hodge module and gives us a cohomology theory which enables us to define the Gopakumar-Vafa invariants mathematically.Comment: Completely rewritten. 94 page

Desingularizations of the moduli space of rank 2 bundles over a curve

Let $X$ be a smooth projective curve of genus $g\ge 3$ and $M_0$ be the moduli space of rank 2 semistable bundles over $X$ with trivial determinant. There are three desingularizations of this singular moduli space constructed by Narasimhan-Ramanan \cite{NR}, Seshadri \cite{Se1} and Kirwan \cite{k5} respectively. The relationship between them has not been understood so far. The purpose of this paper is to show that there is a morphism from Kirwan's desingularization to Seshadri's, which turns out to be the composition of two blow-downs. In doing so, we will show that the singularities of $M_0$ are terminal and the plurigenera are all trivial. As an application, we compute the Betti numbers of the cohomology of Seshadri's desingularization in all degrees. This generalizes the result of \cite{BaSe} which computes the Betti numbers in low degrees. Another application is the computation of the stringy E-function (see \cite{Bat} for definition) of $M_0$ for any genus $g\ge 3$ which generalizes the result of \cite{kiem}

Genotype-based Case-Control Analysis, Violation of Hardy-Weinberg Equilibrium, and Phase Diagrams

We study in detail a particular statistical method in genetic case-control analysis, labeled "genotype-based association", in which the two test results from assuming dominant and recessive model are combined in one optimal output. This method differs both from the allele-based association which artificially doubles the sample size, and the direct chi-square test on 3-by-2 contingency table which may overestimate the degree of freedom. We conclude that the comparative advantage (or disadvantage) of the genotype-based test over the allele-based test mainly depends on two parameters, the allele frequency difference delta and the Hardy-Weinberg disequilibrium coefficient difference delta_epsilon. Six different situations, called "phases", characterized by the two X^2 test statistics in allele-based and genotype-based test, are well separated in the phase diagram parameterized by delta and delta_epsilon. For two major groups of phases, a single parameter theta = tan^-1 (delta/delta_epsilon) is able to achieves an almost perfect phase separation. We also applied the analytic result to several types of disease models. It is shown that for dominant and additive models, genotype-based tests are favored over allele-based tests.Comment: 10 pages, 2 figure

Formal Specification & Analysis of Autonomous Systems in PrCCSL/Simulink Design Verifier

Modeling and analysis of timing constraints is crucial in automotive systems. EAST-ADL is a domain specific architectural language dedicated to safety-critical automotive embedded system design. In most cases, a bounded number of violations of timing constraints in systems would not lead to system failures when the results of the violations are negligible, called Weakly-Hard (WH). We have previously specified EAST-ADL timing constraints in Clock Constraint Specification Language (CCSL) and transformed timed behaviors in CCSL into formal models amenable to model checking. Previous work is extended in this paper by including support for probabilistic analysis of timing constraints in the context of WH: Probabilistic extension of CCSL, called PrCCSL, is defined and the EAST-ADL timing constraints with stochastic properties are specified in PrCCSL. The semantics of the extended constraints in PrCCSL is translated into Proof Objective Models that can be verified using SIMULINK DESIGN VERIFIER. Furthermore, a set of mapping rules is proposed to facilitate guarantee of translation. Our approach is demonstrated on an autonomous traffic sign recognition vehicle case study.Comment: 41 pages, 18 figures, technical report reference of SETTA2018 conferenc

SMT-based Probabilistic Analysis of Timing Constraints in Cyber-Physical Systems

Modeling and analysis of timing constraints is crucial in cyber-physical systems (CPS). EAST-ADL is an architectural language dedicated to safety-critical embedded system design. SIMULINK/STATEFLOW (S/S) is a widely used industrial tool for modeling and analysis of embedded systems. In most cases, a bounded number of violations of timing constraints in systems would not lead to system failures when the results of the violations are negligible, called Weakly-Hard (WH). We have previously defined a probabilistic extension of Clock Constraint Specification Language (CCSL), called PrCCSL, for formal specification of EAST-ADL timing constraints in the context of WH. In this paper, we propose an SMT-based approach for probabilistic analysis of EAST-ADL timing constraints in CPS modeled in S/S: an automatic transformation from S/S models to the input language of SMT solver is provided; timing constraints specified in PrCCSL are encoded into SMT formulas and the probabilistic analysis of timing constraints is reduced to the validity checking of the resulting SMT encodings. Our approach is demonstrated a cooperative automotive system case study.Comment: 2 pages, accepted at FMCAD2018 student foru

Vanishing of the top Chern classes of the moduli of vector bundles

Let $Y$ be a smooth projective curve of genus $g\ge 2$ and let $M_{r,d}(Y)$ be the moduli space of stable vector bundles of rank $r$ and degree $d$ on $Y$. A classical conjecture of Newstead and Ramanan states that $c_i(M_{2,1}(Y))=0$ for $i>2(g-1)$ i.e. the top $2g-1$ Chern classes vanish. The purpose of this paper is to generalize this vanishing result to the rank 3 case by generalizing Gieseker's degeneration method. More precisely, we prove that $c_i(M_{3,1}(Y))=0$ for $i>6g-5$. In other words, the top $3g-3$ Chern classes vanish. Notice that we also have $c_i(M_{3,2}(Y))=0$ for $i>6g-5$