摘要翻译:
我们考虑了一个泊松过程$\eta$在一个具有偏序的可测空间$(\by,\mathcal{Y})上,假设它对于强度度量$\lambda$为$\eta$几乎总是严格的。我们给出了一个Clark-Ocone型公式,它提供了平方可积鞅的显式表示(定义于与$\eta$相关的自然过滤),以前只在特殊情况下知道,当$\lambda$是$\r_+$上的Lebesgue测度和另一个空间$\bx$上的$\sigma$-有限测度的乘积时。我们的证明是新的,并且仅基于泊松过程和随机积分的一些基本性质。我们还考虑了独立随机测度在纯跳跃型意义下的更一般情况,证明了Clark-Ocone型表示导致平方可积鞅的Kunita-Watanabe分解的显式形式。在一个由独立随机测度驱动的相当一般的金融市场中,我们还发现了显式的最小方差套期保值。
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英文标题:
《Martingale representation for Poisson processes with applications to
minimal variance hedging》
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作者:
Guenter Last and Mathew D. Penrose
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最新提交年份:
2010
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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:Computational Finance 计算金融学
分类描述:Computational methods, including Monte Carlo, PDE, lattice and other numerical methods with applications to financial modeling
计算方法,包括蒙特卡罗,偏微分方程,格子和其他数值方法,并应用于金融建模
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英文摘要:
We consider a Poisson process $\eta$ on a measurable space $(\BY,\mathcal{Y})$ equipped with a partial ordering, assumed to be strict almost everwhwere with respect to the intensity measure $\lambda$ of $\eta$. We give a Clark-Ocone type formula providing an explicit representation of square integrable martingales (defined with respect to the natural filtration associated with $\eta$), which was previously known only in the special case, when $\lambda$ is the product of Lebesgue measure on $\R_+$ and a $\sigma$-finite measure on another space $\BX$. Our proof is new and based on only a few basic properties of Poisson processes and stochastic integrals. We also consider the more general case of an independent random measure in the sense of It\^o of pure jump type and show that the Clark-Ocone type representation leads to an explicit version of the Kunita-Watanabe decomposition of square integrable martingales. We also find the explicit minimal variance hedge in a quite general financial market driven by an independent random measure.
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PDF链接:
https://arxiv.org/pdf/1001.3972