Showing papers by "Shige Peng published in 2017"

Mean-field stochastic differential equations and associated PDEs

Abstract: In this paper, we consider a mean-field stochastic differential equation, also called the McKean–Vlasov equation, with initial data (t,x)∈[0,T]×Rd(t,x)∈[0,T]×Rd, whose coefficients depend on both the solution Xt,xsXst,x and its law. By considering square integrable random variables ξξ as initial condition for this equation, we can easily show the flow property of the solution Xt,ξsXst,ξ of this new equation. Associating it with a process Xt,x,PξsXst,x,Pξ which coincides with Xt,ξsXst,ξ, when one substitutes ξξ for xx, but which has the advantage to depend on ξξ only through its law PξPξ, we characterize the function V(t,x,Pξ)=E[Φ(Xt,x,PξT,PXt,ξT)]V(t,x,Pξ)=E[Φ(XTt,x,Pξ,PXTt,ξ)] under appropriate regularity conditions on the coefficients of the stochastic differential equation as the unique classical solution of a nonlocal partial differential equation of mean-field type, involving the first- and the second-order derivatives of VV with respect to its space variable and the probability law. The proof bases heavily on a preliminary study of the first- and second-order derivatives of the solution of the mean-field stochastic differential equation with respect to the probability law and a corresponding Ito formula. In our approach, we use the notion of derivative with respect to a probability measure with finite second moment, introduced by Lions in [Cours au College de France: Theorie des jeu a champs moyens (2013)], and we extend it in a direct way to the second-order derivatives.

Stochastic calculus with respect to G-Brownian motion viewed through rough paths

Abstract: We study rough path properties of stochastic integrals of Ito’s type and Stratonovich’s type with respect to G-Brownian motion. The roughness of G-Brownian motion is estimated and then the pathwise Norris lemma in G-framework is obtained.

Reflected BSDE driven by G-Brownian motion with an upper obstacle

Abstract: In this paper, we study the reflected backward stochastic differential equation driven by G-Brownian motion (reflected G-BSDE for short) with an upper obstacle. The existence is proved by approximation via penalization. By using a variant comparison theorem, we show that the solution we constructed is the largest one.

Stein type characterization for $G$-normal distributions

Abstract: In this article, we provide a Stein type characterization for $G$-normal distributions: Let $\mathcal{N} [\varphi ]=\sup _{\mu \in \Theta }\mu [\varphi ],\ \varphi \in C_{b,Lip}(\mathbb{R} ),$ be a sublinear expectation. $\mathcal{N} $ is $G$-normal if and only if for any $\varphi \in C_b^2(\mathbb{R} )$, we have \[ \int _\mathbb{R} [\frac{x} {2}\varphi '(x)-G(\varphi ''(x))]\mu ^\varphi (dx)=0, \] where $\mu ^\varphi $ is a realization of $\varphi $ associated with $\mathcal{N} $, i.e., $\mu ^\varphi \in \Theta $ and $\mu ^\varphi [\varphi ]=\mathcal{N} [\varphi ]$.

Reflected Solutions of BSDEs Driven by G-Brownian Motion

Abstract: In this paper, we study the reflected solutions of one-dimensional backward stochastic differential equations driven by G-Brownian motion (RGBSDE for short) The reflection keeps the solution above a given stochastic process In order to derive the uniqueness of reflected GBSDEs, we apply a "martingale condition" instead of the Skorohod condition Similar to the classical case, we prove the existence by approximation via penalization

Supermartingale Decomposition Theorem under G-expectation

Abstract: The objective of this paper is to establish the decomposition theorem for supermartingales under the $G$-framework. We first introduce a $g$-nonlinear expectation via a kind of $G$-BSDE and the associated supermartingales. We have shown that this kind of supermartingales have the decomposition similar to the classical case. The main ideas are to apply the uniformly continuous property of $S_G^\beta(0,T)$, the representation of the solution to $G$-BSDE and the approximation method via penalization.