英文标题：
《On small-noise equations with degenerate limiting system arising from
volatility models》
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作者：
Giovanni Conforti, Stefano De Marco, Jean-Dominique Deuschel
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最新提交年份：
2014
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英文摘要：
The one-dimensional SDE with non Lipschitz diffusion coefficient $dX_{t} = b(X_{t})dt + \\sigma X_{t}^{\\gamma} dB_{t}, \\ X_{0}=x, \\ \\gamma<1$ is widely studied in mathematical finance. Several works have proposed asymptotic analysis of densities and implied volatilities in models involving instances of this equation, based on a careful implementation of saddle-point methods and (essentially) the explicit knowledge of Fourier transforms. Recent research on tail asymptotics for heat kernels [J-D. Deuschel, P.~Friz, A.~Jacquier, and S.~Violante. Marginal density expansions for diffusions and stochastic volatility, part II: Applications. 2013, arxiv:1305.6765] suggests to work with the rescaled variable $X^{\\varepsilon}:=\\varepsilon^{1/(1-\\gamma)} X$: while allowing to turn a space asymptotic problem into a small-$\\varepsilon$ problem with fixed terminal point, the process $X^{\\varepsilon}$ satisfies a SDE in Wentzell--Freidlin form (i.e. with driving noise $\\varepsilon dB$). We prove a pathwise large deviation principle for the process $X^{\\varepsilon}$ as $\\varepsilon \\to 0$. As it will become clear, the limiting ODE governing the large deviations admits infinitely many solutions, a non-standard situation in the Wentzell--Freidlin theory. As for applications, the $\\varepsilon$-scaling allows to derive exact log-asymptotics for path functionals of the process: while on the one hand the resulting formulae are confirmed by the CIR-CEV benchmarks, on the other hand the large deviation approach (i) applies to equations with a more general drift term and (ii) potentially opens the way to heat kernel analysis for higher-dimensional diffusions involving such an SDE as a component.
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中文摘要：
非Lipschitz扩散系数为$dX_{t}=b（X_{t}）dt+\\sigma X_{t}^{\\gamma}dB_{t}、\\X_{0}=X、\\\\gamma<1$的一维SDE在数学金融学中被广泛研究。在仔细实施鞍点方法和（本质上）傅立叶变换的明确知识的基础上，有几项工作提出了对涉及该方程实例的模型中的密度和隐含波动率的渐近分析。关于热核尾部渐近性的最新研究[J-D.Deuschel，P.~Friz，A.~Jacquier和S.~Violante.扩散和随机波动的边际密度展开，第二部分：应用。2013年，arxiv:1305.6765]建议使用重新标度的变量$X^{\\varepsilon:=\\varepsilon^{1/（1-\\gamma）}X$：同时允许将空间渐近问题转化为一个小的$\\varepsilon由于固定终点的问题，进程$X^{\\varepsilon}$满足Wentzell--Freidlin形式的SDE（即驱动噪声$\\varepsilon dB$）。我们证明了进程$X^{\\varepsilon}$as$\\varepsilon\\到0$的路径大偏差原理。很明显，控制大偏差的极限常微分方程允许无穷多个解，这是温策尔-弗赖德林理论中的一种非标准情况。对于应用程序，$\\varepsilon$-缩放允许导出过程路径泛函的精确对数渐近性：一方面，所得公式得到CIR-CEV基准的确认，另一方面，大偏差方法（i）适用于具有更一般漂移项的方程，以及（ii）潜在地为以SDE为成分的高维扩散的热核分析开辟了道路。
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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：Quantitative Finance 数量金融学
二级分类：Pricing of Securities 证券定价
分类描述：Valuation and hedging of financial securities, their derivatives, and structured products
金融证券及其衍生产品和结构化产品的估值和套期保值
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