Showing papers by "Shige Peng published in 2022"
TL;DR: This paper transforms the problem into a stochastic Stackelberg differential game (leader-follower problem), then a cross-optimization method (CO method) is developed where the leader’s cost functional and the follower's cost functional are optimized alternatively via deep neural networks.
Abstract: In this paper, we mainly focus on the numerical solution of high-dimensional stochastic optimal control problem driven by fully-coupled forward-backward stochastic differential equations (FBSDEs in short) through deep learning. We first transform the problem into a stochastic Stackelberg differential game(leader-follower problem), then a cross-optimization method (CO method) is developed where the leader’s cost functional and the follower’s cost functional are optimized alternatively via deep neural networks. As for the numerical results, we compute two examples of the investment-consumption problem solved through stochastic recursive utility models, and the results of both examples demonstrate the effectiveness of our proposed algorithm.
2 citations
30 Apr 2022
TL;DR: In this article , the robust limit theorem under moment and consistency conditions was established under a sublinear expectation framework, and a weak convergence approach based on the notions of tightness and weak compactness was developed.
Abstract: This article establishes a universal robust limit theorem under a sublinear expectation framework. Under moment and consistency conditions, we show that, for $\alpha \in(1,2)$, the i.i.d. sequence \[ \left \{ \left( \frac{1}{\sqrt{n}}\sum_{i=1}^{n}X_{i},\frac{1}{n}\sum _{i=1}^{n}Y_{i},\frac{1}{\sqrt[\alpha]{n}}\sum_{i=1}^{n}Z_{i}\right) \right \} _{n=1}^{\infty} \] converges in distribution to $\tilde{L}_{1}$, where $\tilde{L}_{t}=(\tilde {\xi}_{t},\tilde{\eta}_{t},\tilde{\zeta}_{t})$, $t\in [0,1]$, is a multidimensional nonlinear L\'{e}vy process with an uncertainty set $\Theta$ as a set of L\'{e}vy triplets. This nonlinear L\'{e}vy process is characterized by a fully nonlinear and possibly degenerate partial integro-differential equation (PIDE) \[ \left \{ \begin{array} [c]{l} \displaystyle \partial_{t}u(t,x,y,z)-\sup \limits_{(F_{\mu},q,Q)\in \Theta }\left \{ \int_{\mathbb{R}^{d}}\delta_{\lambda}u(t,x,y,z)F_{\mu}(d\lambda)\right. \\ \displaystyle \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left. +\langle D_{y}u(t,x,y,z),q\rangle+\frac{1}{2}tr[D_{x}^{2}u(t,x,y,z)Q]\right \} =0,\\ \displaystyle u(0,x,y,z)=\phi(x,y,z),\ \ \forall(t,x,y,z)\in \lbrack 0,1]\times \mathbb{R}^{3d}, \end{array} \right. \] with $\delta_{\lambda}u(t,x,y,z):=u(t,x,y,z+\lambda)-u(t,x,y,z)-\langle D_{z}u(t,x,y,z),\lambda \rangle$. To construct the limit process $(\tilde{L}_{t})_{t\in \lbrack0,1]}$, we develop a novel weak convergence approach based on the notions of tightness and weak compactness on a sublinear expectation space. We further prove a new type of L\'{e}vy-Khintchine representation formula to characterize $(\tilde{L}_{t})_{t\in [0,1]}$. As a byproduct, we also provide a probabilistic approach to prove the existence of the above fully nonlinear degenerate PIDE.
2 citations
22 Feb 2022
TL;DR: In this article , the maximal distribution on sublinear expectation space and a new type of random fields with the maximally distributed finite-dimensional distribution are introduced, which includes the temporal-spatial situation as a special case.
Abstract: This paper focuses on the maximal distribution on sublinear expectation space and introduces a new type of random fields with the maximally distributed finite-dimensional distribution. The corresponding spatial maximally distributed white noise is constructed, which includes the temporal-spatial situation as a special case due to the symmetrical independence property of maximal distribution. In addition, the stochastic integrals with respect to the spatial or temporalspatial maximally distributed white noises are established in a quite direct way without the usual assumption of adaptability for integrand.
1 citations
TL;DR: In this paper , the authors proposed a method to solve the problem of the problem: the one-dimensional graph.
Abstract:
TL;DR: In this article , the authors introduce two methods under distribution uncertainty, the idea of which is to modify the predicted probability with an additional uncertainty estimation, and the results of experiments show that their methods outperform most of the others in common evaluation metrics, especially the accuracy of the minority class.