arXiv subjects
Publications and source records attributed to Mingshang Hu.
This short note provides a new and simple proof of the convergence rate for Peng's law of large numbers under sublinear expectations, which improves the corresponding results in Song [15] and Fang et al. [3].
We study a stochastic optimal control problem for forward-backward control systems with quadratic generators. In order to establish the first and second-order variational and adjoint equations, we obtain a new estimate for one-dimensional linear BSDEs with unbounded stochastic Lipschitz coefficients involving bounded mean oscillation martingales (BMO-martingales for short) and prove the solvability for a class of multi-dimensional BSDEs with this type. Finally, a new global stochastic maximum principle is deduced.
In this paper, we study a stochastic recursive optimal control problem in which the value functional is defined by the solution of a backward stochastic differential equation (BSDE) under $\tilde{G}$-expectation. Under standard assumptions, we establish the comparison theorem for this kind of BSDE and give a novel and simple method to obtain the dynamic programming principle. Finally, we prove that the value function is the unique viscosity solution of a type of fully nonlinear HJB equation.
In this paper, we study the numerical method for stochastic optimal control problems (SOCPs). By reducing the optimal control problem to the discrete case, we derive a discrete stochastic maximum principle (SMP). With the help of this SMP, we propose an effective discrete recursive method for SOCPs with feedback control. We rigorously analyze errors of the proposed method and prove that the cost obtained by our method is of first-order convergence. Numerical experiments are carried out to support our theoretical results.
In this paper, we consider a stochastic recursive optimal control problem under model uncertainty. In this framework, the cost function is described by solutions of a family of backward stochastic differential equations. With the help of the linearization techniques and weak convergence methods, we derive the corresponding stochastic maximum principle. Moreover, a linear quadratic robust control problem is also studied.
In this paper, we study a class of explicit positive solutions to $G$-heat equations by solving second order nonlinear ordinary differential equations. Based on the positive solutions, we give the sharp order of $G$-capacity $c_σ\left( \left[ -\varepsilon,\varepsilon \right] \right) $ when $\varepsilon \rightarrow0$.
In this paper, we study the existence and uniqueness of viscosity solutions to a kind of Hamilton-Jacobi-Bellman (HJB) equations combined with algebra equations. This HJB equation is related to a stochastic optimal control problem for which the state equation is described by a fully coupled forward-backward stochastic differential equation. By extending Peng's backward semigroup approach to this problem, we obtain the dynamic programming principle and show that the value function is a viscosity solution to this HJB equation. As for the proof of the uniqueness of viscosity solution, the analysis method in Barles, Buckdahn and Pardoux Baeles-BP usually does not work for this fully coupled case. With the help of the uniqueness of the solution to FBSDEs, we propose a novel probabilistic approach to study the uniqueness of the solution to this HJB equation. We obtain that the value function is the minimum viscosity solution to this HJB equation. Especially, when the coefficients are independent of the control variable or the solution is smooth, the value function is the unique viscosity solution.
In this paper, a class of stable explicit $θ$-schemes are proposed for solving anticipated backward stochastic differential equations (anticipated BSDEs) which generator not only contains the present values of the solutions but also the future. We subtly transform the delay process of the generator into the current measurable process, resulting in high-order convergence rate. We also analyze the stability of our numerical schemes and strictly prove the error estimates. Various numerical tests powerful demonstrate high accuracy of the proposed numerical schemes.
This paper is concerned with optimal control of stochastic fully coupled forward-backward linear quadratic (FBLQ) problems with indefinite control weight costs. In order to obtain the state feedback representation of the optimal control, we propose a new decoupling technique and obtain one kind of non-Riccati-type ordinary differential equations (ODEs). By applying the completion-of-squares method, we prove the existence of the solutions for the obtained ODEs under some assumptions and derive the state feedback form of the optimal control. For this FBLQ problem, the optimal control depends on the entire trajectory of the state process. Some sepcial cases are given to illustrate our results.
Hu et. al 2018 studied a stochastic optimal control problem for fully coupled forward-backward stochastic control systems with a nonempty control domain. By assuming a weakly coupled condition, they established an approach to obtain the first-order, second-order variational equations and the adjoint equations for the states X, Y and Z and deduced the global maximum principle. But it is well known that there are several different conditions such as monotonicity condition, weakly coupled condition and other conditions which can guarantee the existence and uniqueness of the solution to fully coupled FBSDEs. In this note, to overcome the limitations of assuming a specific condition, we propose two kinds of assumptions which can guarantee that the approach developed in Hu et. al 2018 is still applicable. Under these two kinds of assumptions, we obtain the global stochastic maximum principle.
We study a stochastic optimal control problem for fully coupled forward-backward stochastic control systems with a nonempty control domain. For our problem, the first-order and second-order variational equations are fully coupled linear FBSDEs. Inspired by Hu (Hu, Probability, Uncertainty and Quantitative Risk, 2(1) (2017):pp 1-20), we develop a new decoupling approach by introducing an adjoint equation which is a quadratic BSDE. By revealing the relations among the terms of the first-order Taylor's expansions, we estimate the orders of them and derive a global stochastic maximum principle which includes a completely new term. Applications to stochastic linear quadratic control problems are investigated.
In this paper, we obtain Lévy's martingale characterization of $G$-Brownian motion without the nondegenerate condition. Base on this characterization, we prove the reflection principle of $G$-Brownian motion. Furthermore, we use Krylov's estimate to get the reflection principle of $\tilde{G}$-Brownian motion.
Within the framework of viscosity solution, we study the relationship between the maximum principle (MP) in [9] and the dynamic programming principle (DPP) in [10] for a fully coupled forward-backward stochastic controlled system (FBSCS) with a nonconvex control domain. For a fully coupled FBSCS, both the corresponding MP and the corresponding Hamilton-Jacobi-Bellman (HJB) equation combine an algebra equation respectively. So this relationship becomes more complicated and almost no work involves this issue. With the help of a new decoupling technique, we obtain the desirable estimates for the fully coupled forward-backward variational equations and establish the relationship. Furthermore, for the smooth case, we discover the connection between the derivatives of the solution to the algebra equation and some terms in the first and second-order adjoint equations. Finally, we study the local case under the monotonicity conditions as in [14,27] and obtain the relationship between the MP in [27] and the DPP in [14].
In this paper we study the stochastic differential equations driven by $G$-Brownian motion ($G$-SDEs for short). We extend the notion of conditional $G$-expectation from deterministic time to the more general optional time situation. Then, via this conditional expectation, we develop the strong Markov property for $G$-SDEs. In particular, we obtain the strong Markov property for $G$-Brownian motion. Some applications including the reflection principle for $G$-Brownian motion are also provided.
The present paper considers a stochastic optimal control problem, in which the cost function is defined through a backward stochastic differential equation with infinite horizon driven by G-Brownian motion. Then we study the regularities of the value function and establish the dynamic programming principle. Moreover, we prove that the value function is the uniqueness viscosity solution of the related HJBI equation.
In this paper, we first use PDE techniques and probabilistic methods to identify a kind of quasi-continuous random variables. Then we give a characterization of the $G$-integrable processes and get a kind of quasi-continuous processes by Krylov's estimates. This result is useful for the development of $G$-stochastic analysis theory. Moreover, it also provides a tool for the study of the non-Markovian Itô processes.
The present paper considers a new kind of backward stochastic differential equations driven by G-Brownian motion, which is called ergodic G-BSDEs. Firstly, the well-posedness of G-BSDEs with infinite horizon is given by a new linearization method. Then, the Feynman-Kac formula for fully nonlinear elliptic partial differential equations is established. Moreover, a new probabilistic approach is introduced to prove the uniqueness of viscosity solution to elliptic PDEs in the whole space. Finally, we obtain the existence of solution to G-EBSDE and some applications are also stated.
In this article, we provide a Stein type characterization for $G$-normal distributions: Let $\mathcal{N}[φ]=\max_{μ\inΘ}μ[φ],\ φ\in C_{b,Lip}(\mathbb{R}),$ be a sublinear expectation. $\mathcal{N}$ is $G$-normal if and only if for any $φ\in C_b^2(\mathbb{R})$, we have \[\int_\mathbb{R}[\frac{x}{2}φ'(x)-G(φ"(x))]μ^φ(dx)=0,\] where $μ^φ$ is a realization of $φ$ associated with $\mathcal{N}$, i.e., $μ^φ\in Θ$ and $μ^φ[φ]=\mathcal{N}[φ]$.