Explicit Solutions of Quadratic FBSDEs Arising From Quadratic Term Structure Models

We provide explicit solutions of certain forward-backward stochastic differential equations (FBSDEs) with quadratic growth. These particular FBSDEs are associated with quadratic term structure models of interest rates and characterize the zero-coupon bond price. The results of this paper are naturally related to similar results on affine term structure models of Hyndman (Math. Financ. Econ. 2(2):107–128, 2009) due to the relationship between quadratic functionals of Gaussian processes and linear functionals of affine processes. Similar to the affine case a sufficient condition for the explicit solutions to hold is the solvability in a fixed interval of Riccati-type ordinary differential equations. However, in contrast to the affine case, these Riccati equations are easily associated with those occurring in linear-quadratic control problems. We also consider quadratic models for a risky asset price and characterize the futures price and forward price of the asset in terms of similar FBSDEs. An example is considered, using an approach based on stochastic flows that is related to the FBSDE approach, to further emphasize the parallels between the affine and quadratic models. An appendix discusses solvability and explicit solutions of the Riccati equations.     

分数维Vasicek利率模型下的欧式期权定价公式

假定股票价格和利率的运动过程服从几何分数维布朗运动,利用风险对冲技术,分数维布朗运动随机分析理论与偏微分方程方法,得到了分数维Vasicek随机利率下欧式期权所满足的定价方程,获得了波动率是对间函数的情形下欧式看涨和看跌期权的一般定价公式以及它们的平价公式.     

关于调和级数的部分和的推导及证明

中世纪后期,数学家Oresme证明了所有调和级数都是发散的,但是调和级数的拉马努金和存在,且为Euler常数.Euler在1734年利用Newton的成果,首先给出了调和级数的部分和的表达式.通过分析Ross,S.M.对经典概率论问题"优惠券收集问题"的解决方法,得到了调和级数的部分和的不同表达式,并运用数学归纳法,变量代换证明了表达式的正确性.     

Zero‐coupon bond prices in the Vasicek and CIR models: Their computation as group‐invariant solutions

We compute prices of zero‐coupon bonds in the Vasicek and Cox–Ingersoll–Ross interest rate models as group‐invariant solutions. Firstly, we determine the symmetries of the valuation partial differential equation that are compatible with the terminal condition and then seek the desired solution among the invariant solutions arising from these symmetries. We also point to other possible studies on these models using the symmetries admitted by the valuation partial differential equations. Copyright © 2007 John Wiley & Sons, Ltd.     

带有优惠券销售的商品利润模型及其求解

根据"满100元返x元"的优惠券的特点,建立了商场商品在保持原来定价的基础上、以商家利润最大化为目的的数学模型;同时,利用遗传算法对此模型优化求解,仿真结果说明该模型及其解法具有一定的实用价值.     

关于公 司债券折价溢价发行的摊销

本文给出计算公司债券售价.各期折价或溢价摊销额的一种新方法.简单实用，误差较小     

On the best case performance of hit and run methods for detecting necessary constraints

The hit and run methods are probabilistic algorithms that can be used to detect necessary (nonredundant) constraints in systems of linear constraints. These methods construct random sequences of lines that pass through the feasible region. These lines intersect the boundary of the region at twohit-points, each identifying a necessary constraint. In order to study the statistical performance of such methods it is assumed that the probabilities of hitting particular constraints are the same for every iteration. An indication of the best case performance of these methods can be determined by minimizing, with respect to the hit probabilities, the expected value of the number of iterations required to detect all necessary constraints. We give a set of isolated strong local minimizers and prove that for two, three and four necessary constraints the set of local minimizers is the complete set of global minimizers. We conjecture that this is also the case for any number of necessary constraints. The results in this paper also apply to sampling problems (e.g., balls from an urn) and to the coupon collector's problem.     

Analytical pricing of defaultable discrete coupon bonds in unified two-factor model of structural and reduced form models

Pricing formulae for defaultable corporate bonds with discrete coupons (under consideration of the government taxes) in the united two-factor model of structural and reduced form models are provided. The aim of this paper is to generalize the two-factor structural model for defaultable corporate discrete coupon bonds (considered in [1]) into the unified model of structural and reduced form models. In our model the bond holders receive the stochastic coupon (which is the discounted value of a predetermined value at the maturity) at predetermined coupon dates and the face value (debt) and the coupon at the maturity as well as the effect of government taxes which are paid on the proceeds of an investment in bonds is considered. The expected default event occurs when the equity value is not sufficient to pay coupon or debt at the coupon dates or maturity and the unexpected default event can occur at the first jump time of a Poisson process with the given default intensity provided by a step function of time variable. We provide the model and pricing formula for equity value and using it calculate expected default barrier. Then we provide pricing model and formula for defaultable corporate bonds with discrete coupons and consider its duration.     

Lie group symmetries as integral transforms of fundamental solutions

We obtain fundamental solutions for PDEs of the form ut=σxγuxx+f(x)ux−μxru by showing that if the symmetry group of the PDE is nontrivial, it contains a standard integral transform of the fundamental solution. We show that in this case, the problem of finding a fundamental solution can be reduced to inverting a Laplace transform or some other classical transform.     

Some bounds on the coupon collector problem

This work addresses on the coupon collector problem and its generalization introduced by Flajolet, Gardy, and Thimonier. In our main results, we show a ratio limit theorem for the random time of the generalized coupon collector problem, and, further, we give the leading term and the geometric rate for the distribution of this random time, when the number of throws is large. For the classical coupon collector problem, we give a bound on the conditional second moment for the number of visits to the coupons, relying strongly on a result of Holst on extremal distributions. © 2004 Wiley Periodicals, Inc. Random Struct. Alg. 2004