An optimal landing problem for a bessel process

Abstract

This type of problem, in which the optimizer controls a stochastic process until a certain event occurs, is known as a homing problem. The above problem can be interpreted as an optimal landing problem, with d representing ground level. Because the parameter 𝜆 is positive, the optimizer tries to reach d as soon as possible, while taking the control costs into account. Therefore, the optimal control 𝑢∗[𝑋(𝑡)] should in general be negative. Moreover, 𝜃 is a risk parameter. If 𝜃 < 1 (respectively, 𝜃 > 1) the optimizer is risk-averse (resp., risk-seeking) and does not want to land too rapidly (resp., wants to land rapidly).