A note on two-sided stochastic control problems (Q1080406)

We consider a class of two-sided stochastic control problems. For each continuous process \(\pi_ t=\pi^+_ t-\pi^-_ t\) with bounded variation, the state process \((x_ t)\) is defined by \[ x_ t=B_ t+\int^{t}_{0}I_{(x_ s\leq -a)}d\pi^+_ s- \int^{t}_{0}I_{(x_ s\geq a)}d\pi^-_ s, \] where a is a positive constant and \((B_ t)\) is a standard Brownian motion. We show the existence of an optimal policy so as to minimize the cost function \[ J(\pi)=E[\int^{\infty}_{0}e^{-\alpha s}x^ 2_ sds], \] with discount rate \(\alpha >0\), associated with \(\pi\).