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stochastic pdes for large portfolios with general mean-reverting volatility processes

Probability, Uncertainty and Quantitative Risk 2024年第3期9卷 263-300页

作者： Ben Hambly Nikolaos Kolliopoulos Mathematical Institute University of OafordRadcliff Observatory QuarterWoodstock RoadOrfordOX26GGUK Department of Mathematics University of Michigan2074 East Hall530 Church StreetAnn ArborMI 48109-1043USA

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.

关键词： stochastic pdes Large portfolios General mean-reverting volatility processes stochastic volatility model Credit risk

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Fully nonlinear stochastic and rough pdes:Classical and viscosity solutions

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Probability, Uncertainty and Quantitative Risk 2020年第1期5卷 154-212页

作者： Rainer Buckdahn Christian Keller Jin Ma Jianfeng Zhang Univ Brest UMR CNRS 6205Laboratoire de Mathematiques de Bretagne AtlantiqueBrest´Franceand Shandong UniversityJinanChina Department of Mathematics University of Central FloridaOrlandoFloridaUnited States Department of Mathematics University of Southern CaliforniaLos AngelesCaliforniaUnited States

We study fully nonlinear second-order(forward)stochastic *** can also be viewed as forward path-dependent pdes and will be treated as rough pdes under a unified *** the most general fully nonlinear case,we develop a local theory of classical solutions and then define viscosity solutions through smooth test *** notion of viscosity solutions is equivalent to the alternative using ***,we prove basic properties such as consistency,stability,and a partial comparison principle in the general *** the diffusion coefficient is semilinear(i.e,linear in the gradient of the solution and nonlinear in the solution;the drift can still be fully nonlinear),we establish a complete theory,including global existence and a comparison principle.

关键词： stochastic pdes path-dependent pdes rough pdes rough paths viscosity solutions comparison principle functional Ito formulaˆ characteristics rough Taylor expansion

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GLOBAL WELL-POSEDNESS OF THE stochastic 2D BOUSSINESQ EQUATIONS WITH PARTIAL VISCOSITY

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Acta Mathematica Scientia 2011年第5期31卷 1968-1984页

作者：蒲学科郭柏灵 College of Mathematics and Statistics Chongqing University Institute of Applied Physics and Computational Mathematics

This paper deals with the stochastic 2D Boussinesq equations with partial viscosity. This is a coupled system of Navier-Stokes/Euler equations and the transport equation for temperature under additive noise. Global we... 详细信息

关键词： stochastic pdes Boussinesq equations global well-posedness

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Mean field limit of a dynamical model of polymer systems

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Science China Mathematics 2013年第12期56卷 2591-2598页

作者： E Weinan SHEN Hao School of Mathematical Sciences and BICMR Peking University Department of Mathematics and PACM Princeton UniversityPrinceton NJ 08544 USA Program in Applied and Computational Mathematics Princeton UniversityPrinceton NJ 08544 USA

This paper provides a mathematically rigorous foundation for self-consistent mean field theory of the polymeric physics. We study a new model for dynamics of mono-polymer systems. Every polymer is regarded as a string of points which are moving randomly as Brownian motions and under elastic forces. Every two points on the same string or on two different strings also interact under a pairwise potential V. The dynamics of the system is described by a system of N coupled stochastic partial differential equations （Spdes）. We show that the mean field limit as N -＋ c~ of the system is a self-consistent McKean-Vlasov type equation, under suitable assumptions on the initial and boundary conditions and regularity of V. We also prove that both the SPDE system of the polymers and the mean field limit equation are well-posed.

关键词： polymers mean field limit stochastic pdes McKean-Vlasov equation

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