BMO martingales and positive solutions of heat equations
Ying Hu Zhongmin Qian
Mathematical Control & Related Fields 2015, 5(3): 453-473 doi: 10.3934/mcrf.2015.5.453
In this paper, we develop a new approach to establish gradient estimates for positive solutions to the heat equation of elliptic or subelliptic operators on Euclidean spaces or on Riemannian manifolds. More precisely, we give some estimates of the gradient of logarithm of a positive solution via the uniform bound of the logarithm of the solution. Moreover, we give a generalized version of Li-Yau's estimate. Our proof is based on the link between PDE and quadratic BSDE. Our method might be useful to study some (nonlinear) PDEs.
keywords: quadratic BSDE. BMO martingale heat equation gradient estimate Li-Yau's estimate
On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions: The critical case
Freddy Delbaen Ying Hu Adrien Richou
Discrete & Continuous Dynamical Systems - A 2015, 35(11): 5273-5283 doi: 10.3934/dcds.2015.35.5273
In F. Delbaen, Y. Hu and A. Richou (Ann. Inst. Henri Poincaré Probab. Stat. 47(2):559--574, 2011), the authors proved that uniqueness of solution to quadratic BSDE with convex generator and unbounded terminal condition holds among solutions whose exponentials are $L^p$ with $p$ bigger than a constant $\gamma$ ($p>\gamma$). In this paper, we consider the critical case: $p=\gamma$. We prove that the uniqueness holds among solutions whose exponentials are $L^\gamma$ under the additional assumption that the generator is strongly convex. These exponential moments are natural as they are given by the existence theorem.
keywords: unbounded terminal conditions Quadratic BSDEs convex generators uniqueness. critical case
Switching game of backward stochastic differential equations and associated system of obliquely reflected backward stochastic differential equations
Ying Hu Shanjian Tang
Discrete & Continuous Dynamical Systems - A 2015, 35(11): 5447-5465 doi: 10.3934/dcds.2015.35.5447
This paper is concerned with the switching game of a one-dimensional backward stochastic differential equation (BSDE). The associated Bellman-Isaacs equation is a system of matrix-valued BSDEs living in a special unbounded convex domain with reflection on the boundary along an oblique direction. In this paper, we show the existence of an adapted solution to this system of BSDEs with oblique reflection by the penalization method, the monotone convergence, and the a priori estimates.
keywords: backward stochastic differential equations oblique reflection. Switching game

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