A finite horizon optimal switching problem with memory and application to controlled SDDEs

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 optimal switching problem for stochastic delay differential equations driven by a Brownian motion and an independent compound Poisson process, when the dynamics of the process depends on the control. 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.We consider a finite horizon problem and thus assume that the terminal time T is fixed with T < ∞.We let (Ω, F, F, P) be a probability space, with F := (F t ) 0≤t≤T a filtration satisfying the usual conditions in addition to being quasi-left continuous.Remark 2.1. Recall here the concepts of quasi-left continuity: A càdlàg process (X t : 0 ≤ t ≤ T ) is quasi-left continuous if for each predictable stopping time γ and every announcing sequence of stopping

Finite Horizon Impulse control of Stochastic Functional Differential Equations

“…We also mention the results of [11] which we extend by considering impulse controls rather than switching controls and a more general trajectory dependence. The results are also related to the work in [12] where an abstract impulse control problem is solved.…”

Non-Markovian Impulse Control Under Nonlinear Expectation

“…This approach was extended to incorporate delivery lag in [17] and, more recently, also to an infinite horizon setting in [10]. A different approach to non-Markovian impulse control was initiated in [25] and then further developed in [18] where interconnected Snell envelopes indexed by controls were used to find solutions to problems with impulsively controlled path-dependent SDEs. We mention also the general formulation of impulse controls in [26], which can be seen as a linear expectation version of the present work, and the work on impulse control of path-dependent SDEs under g-expectation (see [24]) and related systems of backward SDEs (BSDEs) in [28].…”

Management strategies for run-of-river hydropower plants -an optimal switching approach

“…Optimal switching is a relatively new and fast growing field of mathematics combining optimization, SDEs and partial differential equations (PDEs) [5,6,15,17,18,23,25,27,28,29,30,31,32,38,39,40]. However, a literature survey shows that, although the mathematical theory is well developed, applications of optimal switching to real life problems is a far less explored area.…”