An application of reflected diffusions to the problem of choosing between hydro and thermal power generation (Q1208937)

It is shown that a certain type of stochastic control problems has a solution given by a diffusion with vertical reflection on the boundary of a planar set. The system under study is described by the following stochastic differential equation: \[ {{dX_ t} \brack {dK_ t}} = {{\alpha X_ t dt} \brack {dK_ t}} + {{\beta X_ t} \brack 0} dW_ t, \] where \(\alpha>0\), \((W_ t)_{t\geq 0}\) is the standard one- dimensional Brownian motion. The problem is to find a stochastic control \(K_ s\), increasing, which maximizes the following expectation \[ \sup_{K_ s} E \left[ \int_ t^ \infty e^{-rs} ((P_ s D_ s- q(D_ s -K_ s))ds-C(K_ s)dK_ s)\right], \] where \(r>0\), \(q\geq 0\), \(P_ t=\min\{(X_ t/K_ t)^{1/\varepsilon},q\}\) with \(\varepsilon>0\), \(D_ t=P_ t^{-\varepsilon} \cdot X_ t\), and \(C\) a positive continuous nonincreasing real-valued function. The way to solve this problem is via a variational inequality formulation, by first restricting \(K_ s\) to be a Markov control.