Abstract
A stochastic impulse control problem with imperfect controllability of interventions is formulated with an emphasis on applications to ecological and environmental management problems. The imperfectness comes from uncertainties with respect to the magnitude of interventions. Our model is based on a dynamic programming formalism to impulsively control a 1-D diffusion process of a geometric Brownian type. The imperfectness leads to a non-local operator different from the many conventional ones, and evokes a slightly different optimal intervention policy. We give viscosity characterizations of the Hamilton–Jacobi–Bellman Quasi-Variational Inequality (HJBQVI) governing the value function focusing on its numerical computation. Uniqueness and verification results of the HJBQVI are presented and a candidate exact solution is constructed. The HJBQVI is solved with the two different numerical methods, an ordinary differential equation (ODE) based method and a finite difference scheme, demonstrating their consistency. Furthermore, the resulting controlled dynamics are extensively analyzed focusing on a bird population management case from a statistical standpoint.
Highlights
Balancing costs and benefits is essential for successful environmental and ecological management [1, 2]
3.2 Hamilton–Jacobi–Bellman Quasi-Variational Inequality (HJBQVI) and optimal control 3.2.1 Viscosity property Here, we show that the HJBQVI (20) is solvable in a weak sense
Through comparing the computational results between the ordinary differential equation (ODE)-based method and the finite difference scheme, we examine validity of the functional form of the exact solution to the HJBQVI (20) conjectured in the previous section
Summary
Balancing costs and benefits is essential for successful environmental and ecological management [1, 2]. A function φ ∈ LSC[0, +∞) ∩ C(0, +∞) with φ(0) ≥ 0 is a viscosity super-solution to the HJBQVI (20) if the following conditions are satisfied at each x > 0: for any ψ ∈ C[0, +∞) ∩ C2(0, +∞) such that φ – ψ is locally strictly minimized at x with φ(x) – ψ(x) = 0, we have max Lψ + RxM – rxm, φ – Mφ ≥ 0. Remark 3 The value function is the unique viscosity solution to the localized HJBQVI (28)–(29) if – M = 0 and L + RxM – rxm < 0 for sufficiently large x > 0 If this holds true, we can choose a sufficiently large Xmax such that – M = 0 for x > Xmax. The result presented above is consistent with our intuition that the intervention is n (> 1) times less efficient if its impact is expected to be n times smaller than that without the uncertainty
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