Abstract
We consider an optimal switching problem where the terminal reward depends on the entire control trajectory. We show existence of an optimal control by applying a probabilistic technique based on the concept of Snell envelopes. We then apply this result to solve an impulse control problem for stochastic delay differential equations driven by a Brownian motion and an independent compound Poisson process. Furthermore, we show that the studied problem arises naturally when maximizing the revenue from operation of a group of hydro-power plants with hydrological coupling.
Highlights
The standard optimal switching problem is a stochastic optimal control problem of impulse type that arises when an operator controls a dynamical system by switching between the different members in a set of operation modes I = {1, . . . , m}
We show that a special case of the type of switching problems that we consider is that of a controlled stochastic delay differential equation (SDDE), driven by a finite intensity Lévy process
We show that the revenue maximization problem of the hydro-power producer can be formulated as an impulse control problem where the uncertainty is modeled by a controlled SDDE and use our initial result to find an optimal control for this problem
Summary
The standard multi-modes optimal switching problem in finite horizon (T < ∞) can be formulated as finding the control that maximizes. For the case when the underlying uncertainty can be modeled by a diffusion process, generalization to the case when the control enters the drift and volatility term was treated in Elie and Kharroubi (2014) This was further developed to include state constraints in Kharroubi (2016). We show that the problem of maximizing J can be solved under certain assumptions on φ, ψ and the switching costs c·,· by finding an optimal control in terms of a family of interconnected value processes, that we refer to as a verification family. We show that the revenue maximization problem of the hydro-power producer can be formulated as an impulse control problem where the uncertainty is modeled by a controlled SDDE and use our initial result to find an optimal control for this problem.
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