New class of optimal multiple stopping times problems

Abstract

This paper is devoted to study a new discounted nonlinear optimal multiple stopping times problem with discounted factor β > 0 and infinite horizon. Under the right continuity of the underlying process, we show that the problem can be reduced to a sequence of ordinary optimal stopping problems. Next in the Markovian case, we characterise the value function of the problem in terms of β-excessive functions. Finally, in the special case of a diffusion process, we give explicit expressions for the value function of the problem as well as the optimal stopping strategy. As an explicit example in finance, we apply our theoretical results to manage a new generalised swing contract which gives its holder n rights to mark the price X of a stock, where the payment is only allowed at the final exercise time rather than at each exercise time as in the classical swing contact.