论文标题
一个单数的两阶段Stefan问题和粒子通过打击时间相互作用
A singular two-phase Stefan problem and particles interacting through their hitting times
论文作者
论文摘要
我们考虑了一个空间维度的单数两相stefan问题的概率表述，这相当于两个McKean-Vlasov随机微分方程的耦合系统。在系统性风险的财务背景下，该系统将两个竞争区域建模，该区域具有大量相互联系的银行或有违约风险的公司。我们的主要结果表明，存在该解决方案的存在，该解决方案遵守当前问题的自然物理条件。因此，这项工作扩展了在一个空间维度中单一的一相Stefan问题的最新生存结果，该结果可以在[dirt15a]，[ns19a]，[hls18]，[crs20]中找到。在其中，我们的存在结果是通过Skorokhod M1拓扑中有限粒子系统近似的大系统限制获得的。但是，与先前研究的一相情况不同，此处的自由边界不是单调的，因此大型系统限制是通过新颖的论点获得的。
We consider a probabilistic formulation of a singular two-phase Stefan problem in one space dimension, which amounts to a coupled system of two McKean-Vlasov stochastic differential equations. In the financial context of systemic risk, this system models two competing regions with a large number of interconnected banks or firms at risk of default. Our main result shows the existence of a solution whose discontinuities obey the natural physicality condition for the problem at hand. Thus, this work extends the recent series of existence results for singular one-phase Stefan problems in one space dimension that can be found in [DIRT15a], [NS19a], [HLS18], [CRS20]. As therein, our existence result is obtained via a large system limit of a finite particle system approximation in the Skorokhod M1 topology. But, unlike for the previously studied one-phase case, the free boundary herein is not monotone, so that the large system limit is obtained by a novel argument.
