/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

Couplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical monotonicity and Kantorovich duality.- The Wasserstein distances.- Displacement interpolation.- The Monge-Mather shortening principle.- Solution of the Monge problem I: global approach.- Solution of the Monge problem II: Local approach.- The Jacobian equation.- Smoothness.- Qualitative picture.- Optimal transport and Riemannian geometry.- Ricci curvature.- Otto calculus.- Displacement convexity I.- Displacement convexity II.- Volume control.- Density control and local regularity.- Infinitesimal displacement convexity.- Isoperimetric-type inequalities.- Concentration inequalities.- Gradient flows I.- Gradient flows II: Qualitative properties.- Gradient flows III: Functional inequalities.- Synthetic treatment of Ricci curvature.- Analytic and synthetic points of view.- Convergence of metric-measure spaces.- Stability of optimal transport.- Weak Ricci curvature bounds I: Definition and Stability.- Weak Ricci curvature bounds II: Geometric and analytic properties.

Optimal Transport: Old and New

Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.

Stochastic Equations in Infinite Dimensions

Stochastic equations in infinite dimensions

We introduce the Hilbert space-valued Wiener process and the corresponding stochastic integral of Ito type. This is then used together with semigroup theory to obtain existence and uniqueness of weak solutions of linear and semilinear stochastic evolution problems in Hilbert space. Finally, this abstract theory is applied to the linear heat and wave equations driven by additive noise.

/pdf/introduction-to-stochastic-partial-differential-equations-4mpb9c3p8y.pdf

Introduction to stochastic partial differential equations

Introduction: Introduction An overview The general theory of large deviations: Large deviations and exponential tightness Large deviations for stochastic processes Large deviations for Markov processes and semigroup convergence: Large deviations for Markov processes and nonlinear semigroup convergence Large deviations and nonlinear semigroup convergence using viscosity solutions Extensions of viscosity solution methods The Nisio semigroup and a control representation of the rate function Examples of large deviations and the comparison principle: The comparison principle Nearly deterministic processes in $R^d$ Random evolutions Occupation measures Stochastic equations in infinite dimensions Appendix: Operators and convergence in function spaces Variational constants, rate of growth and spectral theory for the semigroup of positive linear operators Spectral properties for discrete and continuous Laplacians Results from mass transport theory Bibliography Index.

Large Deviations for Stochastic Processes

Stochastic scalar conservation laws

In this paper, we study the Heston stochastic volatility model in a regime where the maturity is small but large compared to the mean-reversion time of the stochastic volatility factor. We derive a large deviation principle and compute the rate function by a precise study of the moment generating function and its asymptotic. We then obtain asymptotic prices for out-of-the-money call and put options and their corresponding implied volatilities.

/pdf/short-maturity-asymptotics-for-a-fast-mean-reverting-heston-586nep2rif.pdf

Short-Maturity Asymptotics for a Fast Mean-Reverting Heston Stochastic Volatility Model

In this paper, we study stochastic volatility models in regimes where the maturity is small, but large compared to the mean-reversion time of the stochastic volatility factor. The problem falls in the class of averaging/homogenization problems for nonlinear HJB-type equations where the “fast variable” lives in a noncompact space. We develop a general argument based on viscosity solutions which we apply to the two regimes studied in the paper. We derive a large deviation principle, and we deduce asymptotic prices for out-of-the-money call and put options, and their corresponding implied volatilities. The results of this paper generalize the ones obtained in Feng, Forde and Fouque [SIAM J. Financial Math. 1 (2010) 126–141] by a moment generating function computation in the particular case of the Heston model.

/pdf/small-time-asymptotics-for-fast-mean-reverting-stochastic-1qe66f68hu.pdf

Jin Feng

Papers

Stochastic equations in infinite dimensions

Large Deviations for Stochastic Processes

Stochastic scalar conservation laws

Short-Maturity Asymptotics for a Fast Mean-Reverting Heston Stochastic Volatility Model

Small-time asymptotics for fast mean-reverting stochastic volatility models