We study the mean–variance hedging of an American-type contingent claim that is exercised at a random time in a Markovian setting. This problem is motivated by applications in the areas of employee stock option valuation, credit risk, or equity-linked life insurance policies with an underlying risky asset value guarantee. Our analysis is based on dynamic programming and uses PDE techniques. In particular, we prove that the complete solution to the problem can be expressed in terms of the solution to a system of one quasi-linear parabolic PDE and two linear parabolic PDEs. Using a suitable iterative scheme involving linear parabolic PDEs and Schauder's interior estimates for parabolic PDEs, we show that each of these PDEs has a classical C^{1, 2} solution. Using these results, we express the claim's mean–variance hedging value that we derive as its expected discounted payoff with respect to an equivalent martingale measure that does not coincide with the minimal martingale measure, which, in the context that we consider, identifies with the minimum entropy martingale measure as well as the variance-optimal martingale measure. Furthermore, we present a numerical study that illustrates aspects of our theoretical results.

中文翻译：

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随机到期日的或有债权的均值方差对冲

我们研究了在马尔可夫环境中随机时间行使的美式或然债权的均值-方差对冲。这个问题是由员工股票期权估值、信用风险或具有潜在风险资产价值保证的股票挂钩人寿保险保单领域的应用引起的。我们的分析基于动态规划并使用 PDE 技术。特别是，我们证明了该问题的完整解可以用一个拟线性抛物线偏微分方程组和两个线性抛物线偏微分方程组的解来表示。使用涉及线性抛物型偏微分方程和抛物型偏微分方程 Schauder 内部估计的合适迭代方案，我们表明这些偏微分方程中的每一个都具有经典的C ^{1, 2}解决方案。利用这些结果，我们将索赔的均值-方差对冲价值表示为相对于与最小鞅措施不一致的等效鞅措施的预期贴现收益，在我们考虑的背景下，最小鞅措施与最小熵鞅测度以及方差最优鞅测度。此外，我们提出了一项数值研究来说明我们的理论结果的各个方面。