[Paper Review] Quadratic $G$-BSDEs with convex generators and unbounded terminal conditions

Ying Hu, Shanjian Tang+1 more|arXiv (Cornell University)|Jan 27, 2021

Stochastic processes and financial applications4 citations

TL;DR

This paper establishes the existence and uniqueness of solutions to one-dimensional quadratic $G$-BSDEs with convex (or concave) generators and unbounded terminal conditions by introducing a novel approximation procedure based on the $\theta$-method of Briand and Hu, combined with nonlinear stochastic analysis. It further proves the well-posedness of multi-dimensional $G$-BSDEs with diagonally quadratic generators under exponential moment conditions on the terminal value.

ABSTRACT

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 $θ$-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.

Motivation & Objective

• To address the lack of existence and uniqueness results for one-dimensional quadratic $G$-BSDEs with unbounded terminal values and convex generators under the $G$-expectation framework.
• To extend the classical monotone convergence and approximation techniques to the nonlinear $G$-expectation setting, where standard tools like the monotone convergence theorem do not directly apply.
• To establish well-posedness for multi-dimensional $G$-BSDEs with diagonally quadratic generators under exponential moment conditions on the terminal condition.
• To overcome the challenges posed by the non-deterministic quadratic variation process $\langle B\rangle$ and the presence of non-increasing $G$-martingales $K$ in the $G$-BSDE formulation.
• To generalize existing results on quadratic BSDEs with bounded terminal values to the case of unbounded terminal conditions with arbitrary exponential moments.

Proposed method

• Adapts the $\theta$-method of Briand and Hu [4] to the $G$-BSDE framework to control the growth of the solution and ensure uniqueness under convexity/concavity of the generator.
• Employs a recursive approximation scheme for the $G$-BSDE, constructing a sequence of approximating $G$-BSDEs with bounded terminal values to derive uniform a priori estimates.
• Uses nonlinear stochastic analysis techniques, including BMO-martingale estimates and exponential moment bounds, to control the growth of the solution processes $Y^{(m)}$ and $Z^{(m)}$.
• Applies the extended conditional $G$-expectation and its representation via essential supremum over a set of sublinear measures to handle the non-linear dynamics.
• Establishes uniform bounds on $\mathbb{\hat{E}}[\exp(\gamma \sup_t |Y^{(m)}_t|)]$ for arbitrary $\gamma > 0$, leveraging iterative application of the approximation procedure.
• Uses Hölder’s inequality and iterative refinement of moment estimates to derive uniform integrability and convergence of the approximating sequence to the true solution.

Experimental results

Research questions

• RQ1 Can the existence and uniqueness of solutions to one-dimensional quadratic $G$-BSDEs be established when the terminal condition is unbounded and has exponential moments of arbitrary order?
• RQ2 How can the $\theta$-method be adapted to the $G$-expectation framework to ensure uniqueness under convexity of the generator?
• RQ3 What conditions on the generator and terminal condition guarantee the well-posedness of multi-dimensional $G$-BSDEs with diagonally quadratic structure?
• RQ4 How can the non-deterministic quadratic variation $\langle B\rangle$ and the non-increasing $G$-martingale $K$ be handled in the approximation and a priori estimation process?
• RQ5 Can the classical monotone convergence method be extended to $G$-BSDEs, given the failure of standard monotone convergence under sublinear expectation?

Key findings

• The paper proves the existence and uniqueness of solutions to one-dimensional quadratic $G$-BSDEs with convex (or concave) generators and terminal conditions in $L^p$-spaces with exponential moments of arbitrary order.
• A uniform a priori estimate is derived: $\mathbb{\hat{E}}[\exp(3p\gamma\tilde{\sigma}^2 \sup_t |Y^{(m)}_t|)] \leq |\hat{A}(G)|^{\mu+1} \mathbb{\hat{E}}[\exp(24n(16n)^{\mu-1}p\gamma\tilde{\sigma}^2 |\xi|)]$ for any $p \geq 1$, with $\mu$ iterations.
• The approximation sequence $\{Y^{(m)}, Z^{(m)}, K^{(m)}\}$ is shown to be uniformly bounded in $L^p$-norms for $Y$, $Z$, and $K$, ensuring convergence to a limit solution.
• The solution to the $G$-BSDE is shown to satisfy $\mathbb{\hat{E}}[\exp(\gamma \sup_t |Y_t|)] < \infty$ for all $\gamma > 0$, implying strong integrability under the $G$-expectation.
• For multi-dimensional $G$-BSDEs with diagonally quadratic generators, the well-posedness is established under the same exponential moment condition on the terminal value.
• The method successfully overcomes the lack of standard monotone convergence and Picard iteration in the $G$-framework by combining iterative approximation with moment estimates.