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This paper constructs the assets portfolio of lever corporation by the structural approach. Because irreversibility and uncertainty of corporate bankruptcy, the corporate bankruptcy is equivalent to a default of the bonds. By using the parabolic stochastic partial differential equations (SPDE) which the lookback option satisfied,the assets portfolio pricing model of lever corporation is derived under the mixed jump-diffusion fractional Brownian motion (MJD-fBm) environment. When the lever corporation in the financial crisis, Shareholders use capital injection to make up for operating losses and debt servicing, then the probability of no default before the bonds maturity and the conditional distribution of the lever corporation assets is obtained, and the pricing formula for lookback option is derived. In the end, a numerical example is given to illustrate the influence of different Hurst parameters and risk coefficient and stock asset weight to the default probability of the lever corporation.

In the context of the aging population, longevity risk will increase great economic pressure to the national endowment security system. How to measure and manage longevity risk has become the focus of research in recent years. Based on the Chinese population mortality data, and Lee-Carter model, we introduce DEJD model (double exponential jump diffusion model) to describe the jump asymmetry of time series factors, and prove that DEJD model is more effective than Lee-Carter model in fitting time series factors. In addition, we use the population mortality data predicted by DEJD model to price the SM bonds in Chinese market, providing an important reference for the promotion of SM bond in China.

Modeling analysis and reasonable prediction of medical costs are the basis and foundation for the determination of medical insurance costs. High-dimensional additional information in medical costs plays an important role in long-term prediction. This paper proposes a partial linear multi-indicator additive model to fit and predict longitudinal medical cost data with high-dimensional features and uses two different dimensionality reduction estimation methods to estimate the model and applies the model to a set of high-dimensional dimensions. The longitudinal medical cost data of the variable is used for case analysis.

Prospective phase II trials usually result in failures in phase III trials. For randomized controlled phase II and phase III trials which are conducted with patients randomized to one of two treatments where the variances of the normally distributed responses are assumed to be known, we analytically obtain the estimated and theoretical assurances for the three cases (no, additive, and multiplicative bias adjustments). Under some minor assumptions, we show that the estimated assurances for the three cases are increasing functions of the per group number of patients and the observed treatment effect of the phase II trial, respectively; and for Case 3, the estimated assurance is an increasing function of the retention factor. When the true treatment effect of phase III is assumed to be a known constant, we show that the theoretical assurances for the three cases are constants which are equal to the designed power or one minus the type II error. Moreover, we show that the estimated assurances are always less than the theoretical assurance. We also obtain the analytical formulas of the probabilities of launching a phase III study for the three cases. Moreover, for Case 3, we show that the probability of launching a phase III study is an increasing function of the retention factor. According to our theoretical investigations, we find that the true treatment effect of phase III has no effect in the simulations. Finally, the simulations are conducted to illustrate the theoretical investigations.

In this paper, we introduce three new classes of multivariate risk statistics, named multivariate comonotonic quasiconvex risk statistics, multivariate quasiconvex risk statistics and multivariate empirical-law-invariant quasiconvex risk statistics, respectively. Representation results for them are provided by dual method. The results of this paper is not only the generalization of one-dimensional quasiconvex risk statistics, but also the extension of multivariate convex risk statistics.

We study discrete-time quantum random walks on the $N$-ary tree by a framework for discrete-time quantum random walks, this framework has no need for coin spaces, it just choose the evolution operator with no constraints other than unitarity, and contain path enumeration using regeneration structures and $z$ transform. As a result, we calculate the generating function of the amplitude at the root in closed form.

Lasso is a variable selection method commonly used in machine learning, which is suitable for regression problems with sparsity. Distributed computing is an important way to reduce computing time and improve efficiency when large sample sizes or massive amounts of data are stored on different agents. Based on the equivalent optimization model of Lasso model and the idea of alternating stepwise iteration, this paper constructs a distributed algorithm suitable for Lasso variable selection. And the convergence of the algorithm is also proved. Finally, the distributed algorithm constructed in this paper is compared with cyclic-coordinate descent and ADMM algorithm through numerical experiments. For the sparse regression problem with large sample set, the algorithm proposed in this paper has better advantages in computing time and precision.

In this paper, using the property of NSD random variable sequence, moment inequality and three series theorem, we obtain complete convergence for NSD random variables sequences under certain moment conditions. The results generalize the results of independent sequences for Chow and Lai\ucite{15,16} and Jajte\ucite{17}.

In this paper, we use the complex fourier series expansion method (CFS) to price guaranteed minimum death benefits (GMDB). The main idea is to expand the Fourier series of the auxiliary function. The density function of remaining lifetime has two forms in this paper, namely combination-of-exponentials density and piecewise constant forces of mortality assumption, and the coefficients of series are estimated by using the known characteristic function of the general L\'{e}vy model. We mainly consider the value of GMDB products under call options and put options. In the numerical experiment section, we also demonstrate the advantages of CFS in calculation accuracy and running time by comparing with cosine series expansion method (COS) and Monte Carlo method (MC).

The modes of piecewise and the number of intervalsare particularly important in the formulation of compensation scheme for critical illness insurance in China. In this paper, under the three subsection modes of interval bisection, constant ratio increment and constant ratio decrement, the theoretical models are established with interval quantity as independent variable and serious illness insurance compensation amount as dependent variable. Taking the expected compensation ratio as the standard to measure the compensation level of serious illness insurance, we can get the following results: first, the optimal number of intervals corresponding to the three interval modes are respectively: 3, 3 and 5; second, under the setting of the number of piecewise, the compensation level of the interval equal proportion increasing mode is the highest, the interval equal proportion decreasing mode is the second, and the compensation level of the interval equal proportion increasing mode is the lowest, But there is little difference between the three. Then, based on the data of CHARLS in 2015, we calculated the incidence of family catastrophic medical expenditure under the three interval modes as 7.13%, 7.26% and 7.69% respectively. The result is consistent with that of the theory.

Let \mu_n be the standard Gaussian measure on \mathbb{R}^n and X be a random vector on \mathbb{R}^n with the law \mu_n. U-conjecture states that if f and g are two polynomials on \mathbb{R}^n such that f(X) and g(X) are independent, then there exist an orthogonal transformation Y=LX on \mathbb{R}^n and an integer k such that f\circ L^{-1} is a function of (y_1,y_2,\cdots,y_k) and g\circ L^{-1} is a function of (y_{k+1},y_{k+2},\cdots,y_n). In this case, f and g are said to be unlinked. In this note, we prove that two symmetric, quasi-convex polynomials f and g are unlinked if f(X) and g(X) are independent.

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