论文标题
插入风险度量估算的非反应收敛率
Non-asymptotic convergence rates for the plug-in estimation of risk measures
论文作者
论文摘要
让$ρ$成为一般法律 - 危险风险措施，例如，处于风险的平均值，让$ x $为财务损失，即是一个真正的随机变量。实际上，要么是$ x $的真实分销$μ$是未知的，要么不可能$ρ（μ）$的数值计算。在这两种情况下，要么依靠历史数据或使用蒙特卡洛方法，可以通过有限样本估计器$ρ（μ_n）$（$μ_n$表示$ $ $ $ $ $ $ $）的$ $ $ $ $ $ \ $ $ $样本到大约$ρ（μ）$。在本文中，我们调查了$ρ（μ_n）$至$ρ（μ）$的收敛速率。我们为偏差概率和估计误差的期望提供了非反应收敛速率。分析了这些收敛速率的清晰度。我们的框架进一步允许对冲，我们获得的收敛率既不取决于基础资产的维度，也不取决于可用于交易的期权数量。
Let $ρ$ be a general law--invariant convex risk measure, for instance the average value at risk, and let $X$ be a financial loss, that is, a real random variable. In practice, either the true distribution $μ$ of $X$ is unknown, or the numerical computation of $ρ(μ)$ is not possible. In both cases, either relying on historical data or using a Monte-Carlo approach, one can resort to an i.i.d.\ sample of $μ$ to approximate $ρ(μ)$ by the finite sample estimator $ρ(μ_N)$ (where $μ_N$ denotes the empirical measure of $μ$). In this article we investigate convergence rates of $ρ(μ_N)$ to $ρ(μ)$. We provide non-asymptotic convergence rates for both the deviation probability and the expectation of the estimation error. The sharpness of these convergence rates is analyzed. Our framework further allows for hedging, and the convergence rates we obtain depend neither on the dimension of the underlying assets, nor on the number of options available for trading.
