When analyzing and evaluating risks in insurance, people are often confronted with the situation of incomplete information and insufficient data, which is known as a small-sample problem. In this paper, a one-dimensional small-sample problem in insurance was investigated using the kernel density estimation method （KerM） and general limited information diffusion method （GIDM）. In particular, MacCormack technique was applied to get the solutions of GIDM equations and then the optimal diffusion solution was acquired based on the two optimization principles. Finally, the analysis introduced in this paper was verified by treating some examples and satisfying results were obtained.
In this paper, by using the optimal stopping theory, the semilinear Black-Scholes partial differential equation （PDE） was invesigated in a fixed domain for valuing two assets of American （call-max/put-min） options. From the viscosity solution of a PDE, a unique viscosity solution was obtained for the semilinear Black-Scholes PDE.
The explicit solution to Cauchy problem for linearized system of two-dimensional isentropic flow with axisymmetrical initial data in gas dynamics is given.
In this paper, the Riemann solutions for scalar conservation laws with discontinuous flux function were constructed. The interactions of elementary waves of the conservation laws were concerned, and the numerical simulations were given.
