标题： 泊松过程的鞅表示及其在最小方差对冲中的应用 显示英文标题

标题： Martingale representation for Poisson processes with applications to minimal variance hedging

摘要： 我们考虑一个在可测空间$(\BY,\mathcal{Y})$上的泊松过程$\eta$，该空间配备了一个偏序关系，假设相对于强度测度$\lambda$的$\eta$几乎处处严格。 我们给出一个Clark-Ocone型公式，提供了关于平方可积鞅（相对于与$\eta$相关的自然filtration定义的）的显式表示，这在之前仅知于特殊情况，当$\lambda$是$\R_+$上的Lebesgue测度和另一空间$\BX$上的$\sigma$-有限测度的乘积时。 我们的证明是新的，仅基于泊松过程和随机积分的一些基本性质。 我们还考虑了纯跳跃类型的独立随机测度的更一般情况，并表明Clark-Ocone型表示导致了平方可积鞅的Kunita-Watanabe分解的显式版本。 我们还在由独立随机测度驱动的相当一般的金融市场上找到了显式的最小方差对冲。 显示英文摘要

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.