Optimal implementation of environmental improvement policy with implementation costs (Q2762677)

The author studies, for a geometric Brownian motion, the impulsive control problem NEWLINE\[NEWLINE\begin{aligned} dX_{t} & = \mu X_{t}dt+\sigma X_{t}dw_{t},\quad \tau _{i}\leq t<\tau _{i+1}, \quad i\geq 0,\\ X_{\tau _{i}} &= X_{\tau _{i}^{-}}-\zeta _{i}, \\ X_{0} &= x>0, \end{aligned}NEWLINE\]NEWLINE with \(\mu >0\), \(\sigma >0\), \(w_{t}\) a standard Wiener process, \({\mathcal{F}}_{t}\), \(t\geq 0\), the \(\sigma\)-algebras generated by its past, \(\tau_{0}=0\) and \(\tau_{i}\) the control implementation times (assumed to be \({\mathcal{F}}_{t}\) stopping times, \(0<\tau_{i}\leq \tau_{i+1}\), \(\tau_{i}\rightarrow+\infty\) as \(i\rightarrow+\infty\) a.s.), and \(\zeta_{i}\) control magnitudes (assumed \({\mathcal{F}}_{\tau_{i}}\) measurable). This is a problem related to pollution control, where \(X_{t}\) denotes the pollutant level and \(\zeta_{i}\) its reduction by some environmental improvement action at time \(\tau_{i}\).NEWLINENEWLINENEWLINEThe author assumes that the cost of pollution damage is given by \(D(x)=ax^{2}\) and the cost of improvement is given by \(K(\zeta)=c+b\zeta\), \(a,b,c>0\). The author solves the problem of minimizing the cost function \({\mathbf {E}}\left[ \int_{0}^{+\infty }e^{-rt}D(X_{t})dt+\sum_{r=1}^{+\infty }e^{-r\tau _{i}}K(\zeta _{i})\right] \), where \(r>0\) is the discount rate.