摘要翻译:
设$\mathbb{Q}$和$\mathbb{P}$是等价的概率测度,设$\psi$是随机变量的$j$维向量,从而将$\frac{d\mathbb{Q}}{d\mathbb{P}}$和$\psi$定义为$d$维随机微分方程的弱解$x$。从金融经济学中的内生完备性问题出发,给出了在$\mathbb{Q}$下的每个局部鞅关于$j$维鞅$s_t\set\mathbb{E}^{\mathbb{Q}}[\psi\mathcal{F}_t]$是随机积分的条件。虽然对于$x$的漂移$b=b(t,x)$和波动率$\sigma=\sigma(t,x)$系数只需要具有关于$x$的最小规律性,但它们被假定为关于$t$的解析函数。我们给出了一个反例,说明这个$\sigma$的$T$-分析性假设不能被移除。
---
英文标题:
《Integral representation of martingales motivated by the problem of
endogenous completeness in financial economics》
---
作者:
Dmitry Kramkov (Carnegie Mellon and Oxford) and Silviu Predoiu
(Citigroup)
---
最新提交年份:
2012
---
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
--
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:General Finance 一般财务
分类描述:Development of general quantitative methodologies with applications in finance
通用定量方法的发展及其在金融中的应用
--
---
英文摘要:
Let $\mathbb{Q}$ and $\mathbb{P}$ be equivalent probability measures and let $\psi$ be a $J$-dimensional vector of random variables such that $\frac{d\mathbb{Q}}{d\mathbb{P}}$ and $\psi$ are defined in terms of a weak solution $X$ to a $d$-dimensional stochastic differential equation. Motivated by the problem of \emph{endogenous completeness} in financial economics we present conditions which guarantee that every local martingale under $\mathbb{Q}$ is a stochastic integral with respect to the $J$-dimensional martingale $S_t \set \mathbb{E}^{\mathbb{Q}}[\psi|\mathcal{F}_t]$. While the drift $b=b(t,x)$ and the volatility $\sigma = \sigma(t,x)$ coefficients for $X$ need to have only minimal regularity properties with respect to $x$, they are assumed to be analytic functions with respect to $t$. We provide a counter-example showing that this $t$-analyticity assumption for $\sigma$ cannot be removed.
---
PDF链接:
https://arxiv.org/pdf/1110.3248