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

We consider a structural stochastic volatility model for the loss from a large portfolio of credit risky assets.Both the asset value and the volatility processes are correlated through systemic Brownian motions,with default determined by the asset value reaching a lower boundary.We 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 half-space.The problem is defined in a special weighted Sobolev space.Regularity results are established for solutions to this problem,and then we show that there exists a unique solution.In 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.

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

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 well-posedness result of this system under partial viscosity is proved by using classical energy estimates method.

Fully nonlinear stochastic and rough PDEs:Classical and viscosity solutions

We study fully nonlinear second-order(forward)stochastic PDEs.They can also be viewed as forward path-dependent PDEs and will be treated as rough PDEs under a unified framework.For the most general fully nonlinear case,we develop a local theory of classical solutions and then define viscosity solutions through smooth test functions.Our notion of viscosity solutions is equivalent to the alternative using semi-jets.Next,we prove basic properties such as consistency,stability,and a partial comparison principle in the general setting.If 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.

A Discussion on Two Stochastic Elliptic Modeling Strategies

Based on the study of two commonly used stochastic elliptic models:I:−∇·(a(x,w)·∇u(x,w))=f(x)and II:−∇·(a(x,w)⋄∇u(x,w))=f(x),we constructed a new stochastic elliptic model III:−∇·(a^(−1))^(⋄(−1))⋄∇u(x,w)
=f(x),in[20].The difference between models I and II is twofold:a scaling factor induced by the way of applying theWick product and the regularization induced by theWick product itself.In[20],we showed that model III has the same scaling factor as model I.In this paper we present a detailed discussion about the difference between models I and III with respect to the two characteristic parameters of the random coefficient,i.e.,the standard deviation s and the correlation length lc.Numerical results are presented for both one-and twodimensional cases.

Mean field limit of a dynamical model of polymer systems

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.

Galerkin Method for Wave Equations with Uncertain Coefficients

Polynomial chaos methods(and generalized polynomial chaos methods)have been extensively applied to analyze PDEs that contain uncertainties.However this approach is rarely applied to hyperbolic systems.In this paper we analyze the properties of the resulting deterministic system of equations obtained by stochastic Galerkin projection.We consider a simple model of a scalar wave equation with random wave speed.We show that when uncertainty causes the change of characteristic directions,the resulting deterministic system of equations is a symmetric hyperbolic system with both positive and negative eigenvalues.A consistent method of imposing the boundary conditions is proposed and its convergence is established.Numerical examples are presented to support the analysis.

Error Control in Multi-Element Generalized Polynomial Chaos Method for Elliptic Problems with Random Coefficients

We develop the theory for a robust and efficient adaptive multi-element generalized polynomial chaos(ME-gPC)method for elliptic equations with random coefficients for a moderate number(O(10))of random dimensions.We employ loworder(p≤3)polynomial chaos and refine the solution using adaptivity in the parametric space.We first study the approximation error of ME-gPC and prove its hpconvergence.We subsequently generate local and global a posteriori error estimators.In order to resolve the error equations efficiently,we construct a reduced space using much smaller number of terms in the enhanced polynomial chaos space to capture the errors of ME-gPC approximation.Based on the a posteriori estimators,we propose and implement an adaptive ME-gPC algorithm for elliptic problems with random coefficients.Numerical results for convergence and efficiency are also presented.