This paper investigates a Pareto optimal insurance contract design problem within a behavioral finance framework. In this context, the insured evaluates contracts using the rank-dependent utility (RDU, for short) theory, while the insurer applies the expected value premium principle. The analysis incorporates the incentive compatibility constraint, ensuring that the contracts, called moral-hazard-free, are free from the moral hazard issues identified in Bernard et al. [4]. Initially, the problem is formulated as a non-concave maximization problem involving Choquet expectation. It is then transformed into a quantile optimization problem and addressed using the calculus of variations method. The optimal contracts are characterized by a double-obstacle ordinary differential equation for a semi-linear second-order elliptic operator with nonlocal boundary conditions, which seems new in the financial economics literature. We present a straightforward numerical scheme and a numerical example to compute the optimal contracts. Let 𝜃 and 𝑚₀ represent the relative safety loading and the mass of the potential loss at 0, respectively. We discover that every moral-hazard-free contract is optimal for infinitely many RDU-insured individuals if $0＜\theta ＜\frac{{m}_{0}}{1-{m}_{0}}$. Conversely, certain contracts, such as the full coverage contract, are never optimal for any RDU-insured individual if $\theta >\frac{{m}_{0}}{1-{m}_{0}}$. Additionally, we derive all the Pareto optimal contracts when either the compensation or the retention violates the monotonicity constraint.

This work concerns a type of stochastic systems in which the forward equations are general stochastic differential equations and the backward equations are stochastic variational inequalities. We first prove an averaging principle for general stochastic differential equations in the 𝐿^2𝑝 (𝑝≥1) sense. In addition, a convergence rate for 𝑝=1 is presented. Combining general stochastic differential equations with backward stochastic variational inequalities, we then establish another averaging principle for backward stochastic variational inequalities in the 𝐿² sense using a time discretization method. Finally, we apply our result to nonlinear parabolic partial differential equations to obtain their averaging principles.

This paper investigates the nonparametric estimation of the functional coefficients of the forward-backward stochastic differential equations with random terminal time, focusing on both local constant and local linear estimators. We establish the asymptotic properties of these estimators under both long observation time spans and short sampling intervals, providing precise expressions for the bias and variance terms. Moreover, we propose an empirical likelihood method to construct data-driven confidence intervals for these functional coefficients. We conduct numerical simulations to examine the finite-sample properties of the estimators and to compare the performance of the empirical likelihood method with the conventional approach for constructing confidence intervals based on asymptotic normality.

This paper introduces a novel formulation of mean-field stochastic differential equations driven by G-Brownian motion. The proposed formulation generalises existing approaches within the G-framework and enables the study of Fréchet differentiability. Under non-Lipschitz conditions on the coefficients, we establish the existence and uniqueness of a solution for square-integrable stochastic initial data.

In this study, we develop a theory of optimal stopping problems within the G-expectation framework. To address this problem, we first introduce a type of random times, called G-stopping times, which are specifically suited for this setting. In the discrete-time case with a finite horizon, we define the value function backward and show that it is the smallest G-supermartingale that dominates the payoff process, ensuring the existence of an optimal stopping time. We then extend these results to both the infinite-horizon case and the continuous-time setting. Moreover, we establish the relationship between the value function and the solution of the reflected backward stochastic differential equation driven by G-Brownian motion.

In this study, we investigate the well-posedness of a backward stochastic differential equation with jumps and a central value reflection constraint. The reflection condition is imposed on the real-valued function obtained by solving the equation $\mathbb{E}(\mathrm{arctan}({Y}_{t}-x))=0$ at each time $t\in [0,T]$. The driver depends on the distribution of the solution process $Y$ and follows a general quadratic-exponential structure. The terminal value is assumed to be bounded. Using a fixed-point argument and Bounded Mean Oscillation (BMO in short) martingale theory, we establish the existence and uniqueness of local solutions, which are then extended to construct a global solution over the entire time interval $[0,T]$.