Based on the g-expectation of distributions, we obtain the monotonicity and Jensen’s inequality for the g-expectation of distributions; and for a sequence of distribution functions, we establish a monotone weak convergence theorem, Fatou’s lemma, and a convergence theorem with respect to the g-expectation of distributions.

We study the uniqueness of solutions of backward stochastic differential equations (BSDEs), which generator verifies $ |F(t, y, z)| \leqslant \alpha_{t}+\beta_{t}|y|+\theta_{t}|z|+f(|y|)|z|^{2}$, where $α_t, β_t, θ_t$ are positive processes and the function f is positive, continuous and increasing. The uniqueness of solutions of such BSDEs is derived in two situations, when F is locally Lipschitz and when F is jointly convex. As a byproduct: we show the existence of viscosity solutions to the associated semilinear partial differential equations, which can contain nonlinearity that has quadratic growth in the gradient of the solution.

In this paper, we study the infinite-time mean field games with discounting, establishing an equilibrium where individual optimal strategies collectively regenerate the mean-field distribution. To solve this problem, we partition all agents into a representative player and the social equilibrium. When the optimal strategy of the representative player has the same feedback form as the strategy in the social equilibrium, we say that the system achieves a Nash equilibrium. We construct a Nash equilibrium using the stochastic maximum principle and infinite-time forward-backward stochastic differential equations (FBSDEs). By employing elliptic master equations, a class of distribution-dependent elliptic partial differential equations (PDEs), we provide a representation for the Nash equilibrium strategies. We prove the Yamada − Watanabe type theorem and show weak uniqueness for infinite-time FBSDEs. Furthermore, we prove that the solutions to a system of infinite-time FBSDEs can be employed to construct viscosity solutions for a class of distribution-dependent elliptic PDEs.

The G-expectation is a sublinear expectation. It is an important tool for pricing financial products and managing risk owing to its ability to deal with model uncertainty. The problem is how to efficiently quantify it since the commonly used Monte Carlo method does not work. Fortunately, the expectation of a G-normal random variable can be linked to the viscosity solution of a fully nonlinear G-heat equation. In this paper, we first identify the limits of the uncertainty in the covariance of a two-dimensional G-normal random variable and determine the corresponding G-heat equation. Then, we propose a novel numerical scheme for solving the two-dimensional G-heat equation and pay more attention to the case where there exists uncertainty on the correlation, especially in the case that the correlation ranges from negative to positive. The scheme is monotone, stable, and convergent. The numerical tests show that the scheme is highly efficient.

This paper studies the pricing of contingent claims of American style, using indifference pricing by fully dynamic convex risk measures. We provide a general definition of risk-indifference prices for buyers and sellers in continuous time, in a setting where buyer and seller have potentially different information, and show that these definitions are consistent with no-arbitrage principles. Specifying to stochastic volatility models, we characterize indifference prices via solutions of Backward Stochastic Differential Equations reflected at Backward Stochastic Differential Equations and show that this characterization provides a basis for the implementation of numerical methods using deep learning.

This paper explores the optimal consumption, life insurance, and investment strategies of an individual under the influence of habit formation. We assume that the individual can invest in a risk-free asset, a stock, and an index bond in the financial market, where the stock price follows the 4/2 stochastic volatility model. The primary aim of this paper is to maximize the expected utility of consumption, total bequests, and terminal wealth before retirement or death; the utility of consumption is derived from actual consumption exceeding the established habitual consumption level. By applying the dynamic programming method, we derive the Hamilton-Jacobi-Bellman (HJB) equation that the value function satisfies, obtain the asymptotic solutions for the optimal consumption, life insurance, and investment strategies using the asymptotic expansion method, and prove the corresponding verification theorem. Furthermore, we provide numerical examples to analyze the influence of consumption habit patterns and model parameters on the individual’s optimal strategies.

This paper investigates the optimal investment problem for hybrid pension plans in a financial market with jump-diffusion risky assets, where both the contribution and the benefit are adjusted based on the plan’s performance, and risks are shared across different generations. The investment in a risk-free asset and two risky assets is carried out by the managers of the pension fund. The model of risky asset is assumed to be modulated by a compound Poisson process, with the two risky asset price processes correlated through a common shock. The objective of this study is to seek the optimal investment strategies and risk-sharing arrangements for plan trustees and participants that minimize the costs associated with unstable contribution risks, unstable benefit risks, and discontinuous risks. By applying the stochastic optimal control approach, the closed-form expressions of the optimal strategy and value function are derived. Numerical examples are provided to analyze the effects of parameters on the optimal strategies. In the context of the hybrid pension plan, these strategies effectively facilitate intergenerational risk-sharing.