In this paper, we first study one-dimensional quadratic backward stochastic differential equations driven by $G$-Brownian motions ($G$-BSDEs) with unbounded terminal values. With the help of a $\theta$-method of Briand and Hu [4] and nonlinear stochastic analysis techniques, we propose an approximation procedure to prove existence and uniqueness result when the generator is convex (or concave) and terminal value is of exponential moments of arbitrary order. Finally, we also establish the well-posedness of multi-dimensional G-BSDEs with diagonally quadratic generators.

Quadratic $G$-BSDEs with convex generators and unbounded terminal conditions

In this paper, we consider a class of impulsive stochastic differential equations driven by G-Brownian motion (IGSDEs in short). By means of the G-Lyapunov function method, some criteria on p-th moment stability and p-th moment asymptotical stability for the trivial solutions of IGSDEs are established. An example is presented to illustrate the efficiency of the obtained results.

The p-th moment stability of solutions to impulsive stochastic differential equations driven by G-Brownian motion

We will study the following problem. Let Xt, t ∈ [0, T], be an R^d-valued process defined on a time interval t ∈ [0,T]. Let Y be a random value depending on the trajectory of X. Assume that, at each fixed time t ≤ T, the information available to an agent (an individual, a firm, or even a market) is the trajectory of X before t. Thus at time T , the random value of Y(ω) will become known to this agent. The question is: how will this agent evaluate Y at the time t? We will introduce an evaluation operator et[Y] to define the value of Y given by this agent at time t. This operator et[·] assigns an (Xs)0≤s≤т-dependent random variable Y to an (Xs)0≤s≤t-dependent random variable et[Y]. We will mainly treat the situation in which the process X is a solution of a SDE (see equation (3.1)) with the drift coefficient b and diffusion coefficient a containing an unknown parameter θ = θt. We then consider the so called super evaluation when the agent is a seller of the asset Y. We will prove that such super evaluation is a filtration consistent nonlinear expectation. In some typical situations, we will prove that a filtration consistent nonlinear evaluation dominated by this super evaluation is a g-evaluation. We also consider the corresponding nonlinear Markovian situation.

Filtration Consistent Nonlinear Expectations and Evaluations of Contingent Claims

Dynamic programming principle for stochastic recursive optimal control problem driven by a G-Brownian motion

Quasi-sure exponential stabilization of stochastic systems induced by G-Brownian motion with discrete time feedback control

Quasi-continuous random variables and processes under the G-expectation framework

Stochastic differential equations driven by G-Brownian motion and ordinary differential equations

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 Ito processes.

In this paper, we prove a type of Nagumo theorem on viability properties for stochastic differential equations driven by G-Brownian motion (G-SDEs). In particular, an equivalent criterion is formulated through stochastic contingent and tangent sets. Moreover, by the approach of direct and inverse images for stochastic tangent sets we present checkable conditions which keep the solution of a given G-SDE evolving in some particular sets.

Viability for Stochastic Differential Equations Driven by G-Brownian Motion

The present paper is devoted to the study of the well-posedness of BSDEs with mean reflection whenever the generator has quadratic growth in the $z$ argument. This work is the sequel of Briand et al. [BSDEs with mean reflection, arXiv:1605.06301] in which a notion of BSDEs with mean reflection is developed to tackle the super-hedging problem under running risk management constraints. By the contraction mapping argument, we first prove that the quadratic BSDE with mean reflection admits a unique deterministic flat local solution on a small time interval whenever the terminal value is bounded. Moreover, we build the global solution on the whole time interval by stitching local solutions when the generator is uniformly bounded with respect to the $y$ argument.

Falei Wang

Papers

Quasi-continuous random variables and processes under the G-expectation framework

Stochastic differential equations driven by G-Brownian motion and ordinary differential equations

Quasi-continuous random variables and processes under the G-expectation framework

Viability for Stochastic Differential Equations Driven by G-Brownian Motion

Quadratic BSDEs with mean reflection