Some control problems with random intervention times (Q2749130)

Consider the optimal control problem NEWLINE\[NEWLINEX_t= x+ W_t+ \xi_t,\quad \xi_t= \int_{[0,t]}\theta_s dN_s,NEWLINE\]NEWLINE \((W_t)\) a given Wiener, \((N_s)\) a given Poisson, and \((\theta_s)\) the control process, with cost functional NEWLINE\[NEWLINEE\int_{[0,\infty[} e^{-\alpha t}(X^2_t dt+ cd\xi^*_t)\text{ or } \liminf_{T\to\infty} \int_{[0,T[} (X^2_t dt+ c\xi^*_t),NEWLINE\]NEWLINE where \(\alpha> 0\), \(c> 0\), and NEWLINE\[NEWLINE\xi^*_t= \int_{[0,t]}|\theta_s|dN_s.NEWLINE\]NEWLINE The author obtains an explicit solution \((\theta_t)\). He determines its asymptotic behaviour and the value function, as the intensity of \((N_s)\) tends to infinity.