英文标题：
《Feynman-Kac Formulas for Solutions to Degenerate Elliptic and Parabolic
  Boundary-Value and Obstacle Problems with Dirichlet Boundary Conditions》
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作者：
Paul M.N. Feehan, Ruoting Gong and Jian Song
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最新提交年份：
2015
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英文摘要：
  We prove Feynman-Kac formulas for solutions to elliptic and parabolic boundary value and obstacle problems associated with a general Markov diffusion process. Our diffusion model covers several popular stochastic volatility models, such as the Heston model, the CEV model and the SABR model, which are widely used as asset pricing models in mathematical finance. The generator of this Markov process with killing is a second-order, degenerate, elliptic partial differential operator, where the degeneracy in the operator symbol is proportional to the $2\\alpha$-power of the distance to the boundary of the half-plane, with $\\alpha\\in(0,1]$. Our stochastic representation formulas provide the unique solutions to the elliptic boundary value and obstacle problems, when we seek solutions which are suitably smooth up to the boundary portion $\\Gamma_{0}$ contained in the boundary of the upper half-plane. In the case when the full Dirichlet condition is given, our stochastic representation formulas provide the unique solutions which are not guaranteed to be any more than continuous up to the boundary portion $\\Gamma_{0}$.
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中文摘要：
我们证明了与一般马尔可夫扩散过程有关的椭圆型和抛物型边值问题及障碍问题解的费曼-卡克公式。我们的扩散模型涵盖了几种流行的随机波动率模型，如Heston模型、CEV模型和SABR模型，它们在数学金融中被广泛用作资产定价模型。带killing的马尔可夫过程的生成元是一个二阶退化椭圆偏微分算子，其中算子符号中的简并度与到半平面边界的距离的$2α$-幂成正比，在$\\alpha\\in（0,1]$中，我们的随机表示公式提供了椭圆边值问题和障碍问题的唯一解，当我们寻求适当平滑到边界部分$\\Gamma_{0}$包含在上半平面的边界中。在完全Dirichlet条件给定的情况下，我们的随机表示公式提供了唯一的解，这些解不保证在边界部分$\\Gamma_{0}上是连续的。
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分类信息：

一级分类：Mathematics        数学
二级分类：Probability        概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Mathematics        数学
二级分类：Analysis of PDEs        偏微分方程分析
分类描述：Existence and uniqueness, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDE\'s, conservation laws, qualitative dynamics
存在唯一性，边界条件，线性和非线性算子，稳定性，孤子理论，可积偏微分方程，守恒律，定性动力学
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一级分类：Quantitative Finance        数量金融学
二级分类：Mathematical Finance        数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
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