In this paper, we study the stochastic Volterra integral equation driven by G-Brownian motion (G-SVIE). The existence, uniqueness and two types of continuity of the solution to G-SVIE are obtained. Moreover, combining a new quasilinearization technique with the two-step approximation method, we establish the corresponding comparison theorem for a class of G-SVIEs. In particular, by means of this method, the classical assumptions on partial derivatives of the coefficients are unnecessary.

In this paper, by using the representation theorem for sublinear expectations, we give a simple proof to obtain two inequalities about the sample mean for independent random vectors under sublinear expectations. Finally, we use the main results to study the convergence rate of the multi-dimensional law of large numbers under sublinear expectations and the convergence in capacity.

This paper is devoted to proposing a new asymmetric risk-sensitive criterion involving different risk attitudes toward varying risk sources. The criterion can only be defined through the initial value of the minimal solutions of quadratic backward stochastic differential equations (BSDEs). Before uncovering the mean-variance representation for the introduced criterion by the variational approach, some axioms are given for the first time to characterize a variance decomposition of square integrable random variables. The stochastic control problems under this criterion are described as a kind of stochastic recursive control problems that includes controlled quadratic BSDEs. An asymmetric risk-sensitive global stochastic maximum principle is derived when the quadratic BSDEs are equipped with bounded data. A closed-form solution of a stochastic linear-quadratic risk-sensitive control problem is obtained by introducing a novel completion-of-squares technique for controlled quadratic BSDEs. In addition, a dynamic portfolio optimization problem featuring a stochastic return rate is provided as an application of the asymmetric risk-sensitive control.

In this paper, we study the optimal control problems for stochastic Volterra integral equations driven by G-Brownian motion under Volatility Ambiguity. With the help of G-stochastic analysis techniques and the weak convergence methods, we obtain the variation of the cost functional and the variational inequality. Under the convexity assumptions, we establish the stochastic maximum principle, which serves as both a necessary and sufficient condition for optimal control.

This paper develops a probabilistic approximation scheme for a class of nonstandard, fully nonlinear second-order partial integro-differential equations (PIDEs) associated with nonlinear L & eacute;vy processes under Peng's G-expectation framework. The PIDE features a supremum over a family of a-stable L & eacute;vy measures, possibly degenerate diffusion coefficients, and a non-separable uncertainty set, which places it outside the scope of existing numerical theories for PIDEs. We construct a recursive, piecewise-constant approximation of the viscosity solution and establish explicit error estimates for the scheme. As a key application, our results yield quantitative convergence rates for the universal robust limit theorem under sublinear expectations. This provides a unified treatment of Peng's robust central limit theorem and law of large numbers, as well as the a-stable limit theorem of Bayraktar and Munk, together with explicit Berry-Esseen-type bounds.

In this paper, we obtain a new estimate for uniform integrability under sublinear expectations. Based on this, we establish the limit theorems under nonlinear expectations dominated by sublinear expectations through tightness, and the limit distributions can be completely nonlinear. Finally, we study the limit theorem in a special case, where the limit distribution satisfies positive homogeneity.

In this paper, we study a stochastic optimal control problem under a type of consistent convex expectation dominated by G-expectation. By the separation theorem for convex sets, we get the representation theorems for this convex expectation and conditional convex expectation. Based on these results, we obtain the variational equation for the cost functional by weak convergence and discretization methods. Furthermore, we establish the maximum principle which is sufficient under usual convex assumptions. Finally, we study the linear quadratic control problem by using the obtained maximum principle.

In this paper, we introduce G-Bessel processes for a class of d-dimensional G-Brownian motions. Under the condition of dimensionality d, we obtain that the G-Bessel process is the solution of the stochastic differential equation. Furthermore, under the stricter condition of dimensionality, we establish the existence and uniqueness of a solution of the stochastic differential equation governing the G-Bessel process and prove the nonattainability of the origin for G-Brownian motion.

In this paper, we define the squared G-Bessel process as the square of the modulus of a class of G-Brownian motions and establish that it is the unique solution to a stochastic differential equation. We then derive several path properties of the squared G-Bessel process, which are more profound in the capacity sense. Furthermore, we provide upper and lower bounds for the Laplace transform of the squared G-Bessel process. Finally, we prove that the time-space transformed squared G-Bessel process is a G'-CIR process.

In this paper, we study the relationship between maximum principle (MP) and dynamic programming principle (DPP) for forward-backward control system under consistent convex expectation dominated by G-expectation. Under the smooth assumptions for the value function, we get the relationship between MP and DPP under a reference probability by establishing a useful estimate. If the value function is not smooth, then we obtain the first-order sub-jet and super-jet of the value function at any t. However, the processing method in this case is much more difficult than that when t equals 0.

In this paper, we study the Backward stochastic Volterra integral equation driven by G-Brownian motion (G-BSVIE). By adopting a different backward iteration method, we construct the approximating sequences on each local interval. With the help of G-stochastic analysis techniques and the monotone convergence theorem, the existence, uniqueness, and continuity of the solution over the entire interval are established. Moreover, we derive the comparison theorem.

In this paper, we obtain the existence and uniqueness theorem for backward stochastic differential equation driven by G-Brownian motion (G-BSDE) under degenerate case. Moreover, we propose a new probabilistic method based on the representation theorem of G-expectation and weak convergence to obtain the regularity of fully nonlinear PDE associated to G-BSDE.

We propose a monotone approximation scheme for a class of fully nonlinear degenerate partial integro-differential equations which characterize nonlinear $\alpha$ -stable L & eacute;vy processes under a sublinear expectation space with $\alpha\in(1,2)$ . We further establish the error bounds for the monotone approximation scheme. This in turn yields an explicit Berry-Esseen bound and convergence rate for the $\alpha$ -stable central limit theorem under sublinear expectation.

In this paper, we study the discrete-time approximation schemes for a class of backward stochastic differential equations driven by G-Brownian motion (G-BSDEs) which corresponds to the hedging pricing of European contingent claims. By introducing an auxiliary extended (G) over tilde -expectation space, we propose a class of theta G-schemes to discrete G-BSDEs in this space. With the help of nonlinear stochastic analysis techniques and numerical analysis tools, we prove that our schemes admit half-order convergence for approximating G-BSDE in the general case. In some special cases, our schemes can achieve a first-order convergence rate. Finally, we give an implementable numerical scheme for G-BSDEs based on Peng's central limit theorem and illustrate our convergence results with numerical examples.

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 Lévy process with an uncertainty set $\Theta$ as a set of Lévy triplets. This nonlinear Lévy 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 Lévy-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.

In this paper, we propose a general modeling framework for optimal control of stochastic fully coupled forward-backward linear quadratic (FBLQ) problems with indefinite control weight costs that stem from rational expectations models. We propose a new decoupling technique to obtain the optimal feedback control, which is accompanied by one kind of non-Riccati-type ordinary differential equation (ODE). By applying the completion-of-squares method, we prove the existence of the solutions for the obtained ODEs. The obtained results make it possible to compute the control and value function. For this FBLQ problem, the optimal control should depend on the entire trajectory of the state process. Several examples are given to illustrate our results. Funding: M. Hu’s research was supported by the National Science Foundation (NSF) [Grant 11671231] and the Young Scholars Program of Shandong University [Grant 2016WLJH10]. S. Ji’s research was supported by the NSF [Grant 11571203]. X. Xue’s research was supported by “The Fundamental Research Funds of Shandong University,” the NSF [Grants 12001316 and 61907022], and the Natural Science Foundation of Shandong Province [Grant ZR2019BF015].

In this paper, we study the averaging problem for a class of forward-backward stochastic differential equations driven by G-Brownian motion (G-FBSDEs) with rapidly oscillating coefficients, which corresponds to the singular perturbation problem of a kind of fully nonlinear partial differential equations (PDEs). With the help of the nonlinear stochastic analysis techniques and viscosity solution methods, we prove that the limit distribution of the solution is the unique viscosity solution to a fully nonlinear PDE.

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 backward stochastic differential equations (BSDEs) with unbounded stochastic Lipschitz coefficients involving bounded mean oscillation martingales and prove the solvability for a class of multidimensional 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 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 to a type of fully nonlinear HJB equation.

. In this paper, we study a discrete-time stochastic optimal control problem under distribution uncertainty with convex control domain. By weak convergence method and Sion’s minimax theorem, we obtain the variational inequality for cost functional under a reference probability P ∗ . Moreover, under the square integrability condition for noise and control, we establish the discrete-time stochastic maximum principle under P ∗ . Finally, we introduce a backward algorithm to calculate the reference probability P ∗ and the optimal control u ∗ .