Existence of effort and dividend thresholds under pseudo-exponential discounting

Establish the existence of a solution with effort threshold and dividend threshold satisfying 0<x_1<x_2 for the nonlinear threshold system (equation (\ref{eq2:x1x2})) associated with the effort and dividend problem under the pseudo-exponential discount function \(\beta(t)=(1+\lambda t)e^{-\delta t}\).

Background

For the effort and dividend problem with pseudo-exponential discounting, the candidate equilibrium is characterized by two thresholds: maximal effort is applied below x1x_1, no effort is applied above x1x_1, and dividends are paid once the surplus reaches x2x_2. The thresholds must satisfy the coupled nonlinear system (equation (\ref{eq2:x1x2})) obtained from the smooth-fit condition V′(x1)=−1V'(x_1)=-1 and the condition V′′(x2−)=0V''(x_2-)=0.

The paper derives explicit forms for the auxiliary value functions and proves that, conditional on the existence of thresholds 0<x1<x20<x_1<x_2, the resulting value function is decreasing and convex and the associated control law is an equilibrium. However, unlike the mixture-of-exponentials case, the theoretical existence of such a pair is not established; the authors instead examine it numerically in the subsequent section.

References

In this section, we conduct a numerical analysis for the effort and dividend problem objexmple in the case of the pseudo-exponential discount function eq:pseudo_exp_discount since we have not obtained a result of the existence of the solution $0<x_1<x_2$ to eq2:x1x2 in theory.

eq:pseudo_exp_discount:

β(t):=(1+λt)e−δt, ∀t≥0,\beta(t) := (1+\lambda t) \mathrm{e}^{-\delta t},\ \forall t\ge 0,

eq2:x1x2:

{−λA3μ+σ2θ3(eθ3x1+θ3x1eθ3x1)+λA4−μ+σ2θ4(e−θ4x1−θ4x1e−θ4x1)+B3θ3eθ3x1−B4θ4e−θ4x1=−1,−λA3μ+σ2θ3(2θ3eθ3x2+θ32x2eθ3x2)+λA4−μ+σ2θ4(−2θ4e−θ4x2+θ42x2e−θ4x2)+B3θ32eθ3x2+B4θ42e−θ4x2=0.\left\{\begin{aligned} & -\frac{\lambda A_3}{\mu+\sigma^2\theta_3}\left(\mathrm{e}^{\theta_{3}x_1}+\theta_{3}x_1\mathrm{e}^{\theta_{3}x_1}\right)+\frac{\lambda A_4}{-\mu+\sigma^2\theta_4}\left(\mathrm{e}^{-\theta_{4}x_1}-\theta_{4}x_1\mathrm{e}^{-\theta_{4}x_1}\right)+B_{3} \theta_{3}\mathrm{e}^{\theta_{3} x_1}-B_{4} \theta_{4}\mathrm{e}^{-\theta_{4}x_1}=-1,\\ & -\frac{\lambda A_3}{\mu+\sigma^2\theta_3}\left(2\theta_{3}\mathrm{e}^{\theta_{3}x_2}+\theta_{3}^2x_2\mathrm{e}^{\theta_{3}x_2}\right)+\frac{\lambda A_4}{-\mu+\sigma^2\theta_4}\left(-2\theta_{4}\mathrm{e}^{-\theta_{4}x_2}+\theta_{4}^2x_2\mathrm{e}^{-\theta_{4}x_2}\right)+B_{3} \theta_{3}^2\mathrm{e}^{\theta_{3} x_2}+B_{4} \theta_{4}^2\mathrm{e}^{-\theta_{4}x_2} = 0. \end{aligned}\right.

— Equilibria for Time-inconsistent Regular-singular Control Problems  (2609.08877 - Jia et al., 8 Sep 2026) in Section 5, subsection “Pseudo-exponential discount function,” immediately before Section 6, “Numerical analysis”