Probabilistic aspects of finite-fuel, reflected follower problems

Abstract

The issue is that of following thepath of a Brownian particle by a process of bounded total variation and subject to a reflecting barrier at the origin, in such a way as to minimize expected total cost over a finite horizon. We establish the existence of optimal processes and the dynamic programming equations for this question, and show (by purely probabilistic arguments) its relation to an appropriate family of optimal stopping problems with absorption at the origin.

Optimal Control under a Dynamic Fuel Constraint

“…More specifically, Theorem 1 shows that an optimal control policy will exercise control whenever its impact is maximal as measured by the Snell envelope of the cost functional's subgradient at the optimum; it also shows that actually all available fuel should be spent whenever this Snell envelope tends to decrease. The occurrence of Snell envelopes in this characterization highlights the intimate relationship between singular control and optimal stopping problems which has already been observed in Karatzas andShreve (1984, 1985) or El Karoui and Karatzas (1988Karatzas ( , 1991. The construction of an optimal policy is achieved in Theorem 2 which relates the dynamic finite fuel problem with a stochastic representation theorem obtained in Bank and El Karoui (2004).…”

Connections between Singular Control and Optimal Switching

“…Given any z ∈ I such that conditions (16) and (17) hold, the value of the Dynkin game exists and is equal to…”

Variational inequalities and the pricing of American options

“…Equality (2.3) is essentially the expression of the Snell envelope in terms of the 'reduite' (see [12]). It can bc proved directly, by arguing that the sup in (2.2) and the essential sup in (2.3) are the same when ~-t.r is replaced by the set of stopping times with respect to the filtration (~t.~)~t of the increments W~ -Wt, s/> t (see [213 for details).…”