英文标题：
《Heat Kernels, Solvable Lie Groups, and the Mean Reverting SABR
  Stochastic Volatility Model》
---
作者：
Siyan Zhang, Anna L. Mazzucato, Victor Nistor
---
最新提交年份：
2016
---
英文摘要：
  We use commutator techniques and calculations in solvable Lie groups to investigate certain evolution Partial Differential Equations (PDEs for short) that arise in the study of stochastic volatility models for pricing contingent claims on risky assets. In particular, by restricting to domains of bounded volatility, we establish the existence of the semi-groups generated by the spatial part of the operators in these models, concentrating on those arising in the so-called \"SABR stochastic volatility model with mean reversion.\" The main goal of this work is to approximate the solutions of the Cauchy problem for the SABR PDE with mean reversion, a parabolic problem the generator of which is denoted by $L$. The fundamental solution for this problem is not known in closed form. We obtain an approximate solution by performing an expansion in the so-called volvol or volatility of the volatility, which leads us to study a degenerate elliptic operator $L_0$, corresponding the the zero-volvol case of the SABR model with mean reversion, to which the classical results do not apply. However, using Lie algebra techniques we are able to derive an exact formula for the solution operator of the PDE $\\partial_t u - L_0 u = 0$. We then compare the semi-group generated by $L$--the existence of which does follows from standard arguments--to that generated by $L_0$, thus establishing a perturbation result that is useful for numerical methods for the SABR PDE with mean reversion. In the process, we are led to study semigroups arising from both a strongly parabolic and a hyperbolic problem.
---
中文摘要：
我们使用交换子技术和可解李群中的计算来研究在风险资产未定权益定价的随机波动率模型研究中出现的某些演化偏微分方程（简称PDE）。特别是，通过限制波动率有界的区域，我们确定了这些模型中算子的空间部分生成的半群的存在性，重点关注了所谓的“具有均值回归的SABR随机波动率模型”中产生的半群这项工作的主要目标是近似求解具有均值回复的SABR偏微分方程的柯西问题，这是一个抛物问题，其生成元用$L$表示。这个问题的根本解决方案是封闭的。通过对所谓的volvol或波动率进行展开，我们得到了一个近似解，这导致我们研究了退化椭圆算子$L_0$，对应于具有均值回复的SABR模型的零volvol情形，经典结果不适用于该情形。然而，使用李代数技术，我们能够导出PDE$\\partial_t u-L_0 u=0$的解算子的精确公式。然后，我们将由$L$生成的半群与由$L_0$生成的半群进行比较，从而得出一个微扰结果，该结果对于具有均值回复的SABR偏微分方程的数值方法非常有用。在此过程中，我们将研究由强抛物和双曲问题产生的半群。
---
分类信息：

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
--
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
--

---
PDF下载：
-->