Stochastic PDEs for large portfolios with general mean-reverting volatility processes

作     者：Ben Hambly Nikolaos Kolliopoulos 

作者机构：Mathematical InstituteUniversity of OafordRadcliff Observatory QuarterWoodstock RoadOrfordOX26GGUK Department of MathematicsUniversity of Michigan2074 East Hall530 Church StreetAnn ArborMI 48109-1043USA 

出 版 物：《Probability, Uncertainty and Quantitative Risk》 (概率、不确定性与定量风险（英文）)

年 卷 期：2024年第9卷第3期

页      面：263-300页

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：supported financially by the United Kingdom Engineering and Physical Sciences Research Council (Grant No.EP/L015811/1) by the Foundation for Education and European Culture (founded by Nicos&Lydia Tricha) 

主　　题：Stochastic PDEs Large portfolios General mean-reverting volatility processes Stochastic volatility model Credit risk 

摘      要：We consider a structural stochastic volatility model for the loss from a large portfolio of credit risky *** the asset value and the volatility processes are correlated through systemic Brownian motions,with default determined by the asset value reaching a lower *** prove that if our volatility models are picked from a class of mean-reverting diffusions,the system converges as the portfolio becomes large and,when the vol-of-vol function satisfies certain regularity and boundedness conditions,the limit of the empirical measure process has a density given in terms of a solution to a stochastic initial-boundary value problem on a *** problem is defined in a special weighted Sobolev *** results are established for solutions to this problem,and then we show that there exists a unique *** contrast to the CIR volatility setting covered by the existing literature,our results hold even when the systemic Brownian motions are taken to be correlated.