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Mingshang Hu

Publications and source records attributed to Mingshang Hu.

54 records · Page 3Linked to original sources
arXiv2016

Stein Type Characterization for $G$-normal Distributions

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 $μ^φ[φ]=\mathcal{N}[φ]$.

math.PR↗
arXiv2015

Stochastic maximum principle for stochastic recursive optimal control problem under volatility ambiguity

We study a stochastic recursive optimal control problem in which the cost functional is described by the solution of a backward stochastic differential equation driven by G-Brownian motion. Some of the economic and financial optimization problems with volatility ambiguity can be formulated as such problems. Different from the classical variational approach, we establish the maximum principle by the linearization and weak convergence methods.

math.OC↗
arXiv2014

Dynamic Programming Principle for Stochastic Recursive Optimal Control Problem under G-framework

In this paper, we study a stochastic recursive optimal control problem in which the cost functional is described by the solution of a backward stochastic differential equation driven by G-Brownian motion. Under standard assumptions, we establish the dynamic programming principle and the related fully nonlinear HJB equation in the framework of G-expectation. Finally, we show that the value function is the viscosity solution of the obtained HJB equation.

math.OC↗
arXiv2014

Invariant and ergodic measures for G-diffusion processes

In this paper we study the problems of invariant and ergodic measures under G-expectation framework. In particular, the stochastic differential equations driven by G-Brownian motion have the unique invariant and ergodic measures. Moreover, the invariant and ergodic measures of G-SDEs are also sublinear expectations. However, the invariant measures may not coincide with ergodic measures, which is different from the classical case.

math.PR↗
arXiv2013

Extended Conditional G-Expectations and Related Stopping Times

In this paper we extend the definition of time conditional G-expectations $\mathbb{\hat{E}}_{t}[\cdot]$ to a larger domain on which the dynamical consistency still holds. In fact we can consistently define, by taking the limit, the time conditional expectations for each random variable $X$ which is the downward limit (resp. upward limit) of a monotone sequence $\{X_{i}\}$ in $L_{G}^{1}(Ω)$. To accomplish this procedure, some careful analysis is needed. Moreover, we give a suitable definition of stopping times and obtain the optional stopping theorem. We also provide some basic and interesting properties for the extended conditional G-expectations.

math.PR↗
arXiv2013

A stochastic recursive optimal control problem under the G-expectation framework

In this paper, we study a stochastic recursive optimal control problem in which the objective functional is described by the solution of a backward stochastic differential equation driven by G-Brownian motion. Under standard assumptions, we establish the dynamic programming principle and the related Hamilton-Jacobi-Bellman (HJB) equation in the framework of G-expectation. Finally, we show that the value function is the viscosity solution of the obtained HJB equation.

math.OC↗
arXiv2013

A note on pricing of contingent claims under G-expectation

In this paper, we study the pricing of contingent claims under G-expectation. In order to accomodate volatility uncertainty, the price of the risky security is supposed to governed by a general linear stochastic differential equation (SDE) driven by G-Brownian motion. Utilizing the recently developed results of Backward SDE driven by G-Brownian motion, we obtain the superhedging and suberhedging prices of a given contingent claim. Explicit results in the Markovian case are also derived.

math.PR↗
arXiv2012

Backward Stochastic Differential Equations Driven by G-Brownian Motion

In this paper, we study backward stochastic differential equations driven by a G-Brownian motion. The solution of such new type of BSDE is a triple (Y,Z,K) where K is a decreasing G-martingale. Under a Lipschitz condition for generator f and g in Y and Z. The existence and uniqueness of the solution (Y,Z,K) is proved. Although the methods used in the proof and the related estimates are quite different from the classical proof for BSDEs, stochastic calculus in G-framework plays a central role.

math.PR↗
arXiv2011

The Independence under Sublinear Expectations

We show that, for two non-trivial random variables X and Y under a sublinear expectation space, if X is independent from Y and Y is independent from X, then X and Y must be maximally distributed.

math.PR↗
arXiv2010

Function spaces and capacity related to a Sublinear Expectation: application to G-Brownian Motion Pathes

In this paper we give some basic and important properties of several typical Banach spaces of functions of $G$-Brownian motion pathes induced by a sublinear expectation--G-expectation. Many results can be also applied to more general situations. A generalized version of Kolmogorov's criterion for continuous modification of a stochastic process is also obtained. The results can be applied to continuous time dynamic and coherent risk measures in finance in particular for path-dependence risky positions under situations of volatility model uncertainty.

math.PR↗
arXiv2009

G-Lévy Processes under Sublinear Expectations

We introduce G-Lévy processes which develop the theory of processes with independent and stationary increments under the framework of sublinear expectations. We then obtain the Lévy-Khintchine formula and the existence for G-Lévy processes. We also introduce G-Poisson processes.

math.PR↗
arXiv2009

On the Representation Theorem of G-Expectations and Paths of G--Brownian Motion

We give a very simple and elementary proof of the existence of a weakly compact family of probability measures $\{P_θ:θ\in Θ\}$ to represent an important sublinear expectation--G-expectation $\mathbb{E}[\cdot]$. We also give a concrete approximation of a bounded continuous function $X(ω)$ by an increasing sequence of cylinder functions $L_{ip}(Ω)$ in order to prove that $C_{b}(Ω)$ belongs to the $\mathbb{E}[|\cdot|]$-completion of the $L_{ip}(Ω)$.

math.PR↗