We consider an asymptotic SPDE description of a large portfolio model where the underlying asset prices evolve according to certain stochastic volatility models with default upon hitting a lower barrier. The asset prices and their volatilities are correlated through systemic Brownian motions, and the SPDE is obtained on the positive half-space along with a Dirichlet boundary condition. We study the convergence of the loss from the system, which is given in terms of the total mass of a solution to our stochastic initial-boundary value problem, under fast mean-reversion of the volatility. We consider two cases. In the first case, the volatilities are sped up towards a limiting distribution and the system converges only in a weak sense. On the other hand, when only the mean-reversion coefficients of the volatilities are allowed to grow large, we see a stronger form of convergence of the system to its limit. Our results show that in a fast mean-reverting volatility environment, we can accurately estimate the distribution of the loss from a large portfolio by using an approximate constant volatility model which is easier to handle.

中文翻译：

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大型随机波动率模型的快速均值回归渐近性

我们考虑大型投资组合模型的渐近SPDE描述，其中基础资产价格根据某些随机波动率模型演变，并在遇到较低的壁垒时出现违约。资产价格及其波动率通过系统的布朗运动进行关联，SPDE是在正半空间以及Dirichlet边界条件下获得的。我们研究了系统损失的收敛性，它是在波动的快速均值回复下，根据我们的随机初始边界值问题的解决方案的总质量给出的。我们考虑两种情况。在第一种情况下，波动率会加速趋向极限分布，并且系统只会在较弱的意义上收敛。另一方面，当仅允许波动率的均值回归系数增大时，我们看到了一种更强大的系统融合形式，可以发挥其最大作用。我们的结果表明，在快速的均值回复波动率环境中，我们可以使用易于处理的近似恒定波动率模型来准确估算大型投资组合的损失分布。