PRICING CATASTROPHE OPTIONS WITH COUNTERPARTY CREDIT RISK IN A REDUCED FORM MODEL

In this paper, we study the price of catastrophe Options with counterparty credit risk in a reduced form model. We assume that the loss process is generated by a doubly stochastic Poisson process, the share price process is modeled through a jump-diffusion process which is correlated to the loss process, the interest rate process and the default intensity process are modeled through the Vasicek model： We derive the closed form formulae for pricing catastrophe options in a reduced form model. Furthermore, we make some numerical analysis on the explicit formulae.

FLIM as a Promising Tool for Cancer Diagnosis and Treatment Monitoring

Fluorescence lifetime imaging microscopy(FLIM)has been rapidly developed over the past 30 years and widely applied in biomedical engineering.Recent progress in fluorophore-dyed probe design has widened the application prospects of fluorescence.Because fluorescence lifetime is sensitive to microenvironments and molecule alterations,FLIM is promising for the detection of pathological conditions.Current cancer-related FLIM applications can be divided into three main categories:(i)FLIM with autofluorescence molecules in or out of a cell,especially with reduced form of nicotinamide adenine dinucleotide,and flavin adenine dinucleotide for cellular metabolism research;(ii)FLIM with Förster resonance energy transfer for monitoring protein interactions;and(iii)FLIM with fluorophore-dyed probes for specific aberration detection.Advancements in nanomaterial production and efficient calculation systems,as well as novel cancer biomarker discoveries,have promoted FLIM optimization,offering more opportunities for medical research and applications to cancer diagnosis and treatment monitoring.This review summarizes cutting-edge researches from 2015 to 2020 on cancer-related FLIM applications and the potential of FLIM for future cancer diagnosis methods and anti-cancer therapy development.We also highlight current challenges and provide perspectives for further investigation.

On Quasi-Reduced Quadratic Forms

With the help of continued fractions, we plan to list all the elements of the set Q△ = {aX2 ＋ bXY ＋ cY2 ： a,b, c ∈Z, b2 - 4ac = △ with 0 ≤ b 〈 √△}of quasi-reduced quadratic forms of fundamental discriminant △. As a matter of fact, we show that for each reduced quadratic form f = aX2 ＋ bXY ＋ cY2 = （a, b, c） of discriminant △〉0（and of sign σ（f） equal to the sign of a）, the quadratic forms associated with f and defined by {〈a＋bu＋cu2,b＋2cu.c〉},with 1≤σ（f）u≤b/2｜c｜ （whenever they exist）, 〈c,-b-2cu,a＋bu＋cu2〉 with b/2｜c｜≤σ（f）u≤[w（f）]=[b＋√△/2｜c｜], are all different from one another and build a set I（f） whose cardinality is #I（f）={1＋[ω（f）],when（2c）｜b,[ω（f）],when （2c）｜b. If f and g are two different reduced quadratic forms, we show that I（f） ∩ I（g） = Ф. Our main result is that the set Q△ is given by the disjoint union of all I（f） with f running through the set of reduced quadratic forms of discriminant △〉0. This allows us to deduce a formula for #（Q△） involving sums of partial quotients of certain continued fractions.

The valuation of multi-counterparties CDS with credit rating migration

In this paper,the pricing of a Credit Default Swap(CDS)contract with multiple counterparties is considered.The pricing model takes into account the credit rating migration risk of the reference.It is a new model established under the reduced form framework,where the intensity rates are assumed to have structural styles.We derive from it a non-linear partial differential equation system where both positive and negative correlations of counterparties and the references are considered via a single factor model.Then,an ADI(Alternating Direction Implicit)difference method is used to solve the partial differential equations by iteration.From the numerical results,the comparison of multi-counterparty CDS contract and the standard one are analyzed respectively.Moreover,the impact of default parameters on value of the contracts are discussed.

On Non-Decomposable Hermitian Forms over Gaussian Domain

Methods are presented for the construction of non-decomposable Hermitian forms over the ring of integers of an imaginary quadratic field.

Accurate prediction of air quality response to emissions for effective control policy design

Designing effective control policy requires accurate quantification of the relationship between the ambient concentrations of O3and PM2.5and the emissions of their precursors.However,the challenge is that precursor reduction does not necessarily lead to decreases in the concentrations of O3and PM2.5,which are formed by multiple precursors under complex physical and chemical processes;this calls for the development of advanced model technologies to provide accurate predictions of the nonlinear responses of air quality to emissions.Different from the traditional sensitivity analysis and source apportionment methods,the reduced form models(RFMs)based on chemical transport models(CTMs)are able to quantify air quality responses to emissions more accurately and efficiently with lower computational cost.Here we review recent approaches used in RFMs and compare their structures,advantages and disadvantages,performance and applications.In general,RFMs are classified into three types including(1)sensitivity-based models,(2)models with simplified chemistry and physical processes,and(3)statistical models,with considerable differences in principles,characteristics and application ranges.The prediction of nonlinear responses by RFMs enables more in-depth analysis,not only in terms of real-time prediction of concentrations and quantification of human exposure,health impacts and economic damage,but also in optimizing control policies.Notably,data assimilation and emission inventory inversion based on the nonlinear response of concentrations to emissions can also be greatly beneficial to air pollution control management.In future studies,improvement in the performance of CTMs is exceedingly crucial to obtain a more reliable baseline for the prediction of air quality responses.Development of models to determine the air quality response to emissions under varying meteorological conditions is also necessary in the context of future climate changes,which pose great challenges to the quantification of response relationships.Additionally,with rising requirements for fine-scale air quality management,improving the performance of urban-scale simulations is worth considering.In short,accurate predictions of the response of air quality to emissions,though challenging,holds great promise for the present as well as for future scenarios.

Multiple-order rogue wave solutions to a(2+1)-dimensional Boussinesq type equation

In this paper,based on the Hirota bilinear method and symbolic computation approach,multipleorder rogue waves of(2+1)-dimensional Boussinesq type equation are constructed.The reduced bilinear form of the equation is deduced by the transformation of variables.Three kinds of rogue wave solutions are derived by means of bilinear equation.The maximum and minimum values of the first-order rogue wave solution are given at a specific moment.Furthermore,the second-order and third-order rogue waves are explicitly derived.The dynamic characteristics of three kinds of rogue wave solutions are shown by three-dimensional plot.