Optimal Control under a Dynamic Fuel Constraint

Abstract

We present a new approach to solve optimal control problems of the monotone follower type. The key feature of our approach is that it allows to include an arbitrary dynamic fuel constraint. Instead of dynamic programming, we use the convexity of our cost functional to derive a first order characterization of optimal policies based on the Snell envelope of the objective functional's gradient at the optimum. The optimal control policy is constructed explicitly in terms of the solution to a representation theorem for stochastic processes obtained in Bank and El Karoui (2004). As an illustration, we show how our methodology allows to extend the scope of the explicit solutions obtained for the classical monotone follower problem and for an irreversible investment problem arising in economics.AMS 2000 subject classification. 49J55, 93E20, 60H30, 91B28

Optimal Boundary Surface for Irreversible Investment with Stochastic Costs

“…The introduction of a stochastic investment cost Y t is very natural from the point of view of economic modelling (see e.g. [5]); nevertheless, it makes the analysis of the optimal boundary of (1.1) rather difficult.…”

“…Among others, these include dynamic programming techniques (see, e.g., [23], [29], [35] and [38]), stochastic first-order conditions and the Bank-El Karoui's Representation Theorem [4] (see, e.g., [5], [15], [24] and [44]), the transformation method of [7] in the case of one-dimensional problems, and the analytical study of non linear PDEs with gradient constraints (see for example [46] and [47]). The introduction of a stochastic investment cost Y t is very natural from the point of view of economic modelling (see e.g.…”

A Class of Singular Control Problems and the Smooth Fit Principle

“…Mathematical analysis of such control problems has evolved considerably from the initial heuristics to the more sophisticated and standard stochastic control approach, and from the very special case study to general payoff functions. (See Harrison and Taksar (1983); Karatzas (1985); Karatzas and Shreve (1985); El Karoui and Karatzas ( , 1989; Ma (1992); Zervos (1994, 1998); Boetius and Kohlmann (1998); Alvarez (2000Alvarez ( , 2001; Bank (2005); Boetius (2005)). Most recently, Merhi and Zervos (2007) analyzed this problem in great generality and provided explicit solutions for the special case where the payoff is of Cobb-Douglas type.…”

Solving Singular Control from Optimal Switching

“…In mathematical economics, a typical (ir)reversible investment problem can be formulated as a singular control problem in which a company, by adjusting its production capacity through expansion and contraction according to market fluctuations, wishes to maximize its overall expected net profit over an infinite horizon. This problem has been investigated by numerous authors (See for instance Davis et al (1987); Kobila (1993); Abel and Eberly (1997); Baldursson and Karatzas (1997); Øksendal (2000); Scheinkman and Zariphopoulou (2001); Wang (2003); Chiarolla and Haussmann (2005); Bank (2005); Guo and Pham (2005), and Merhi and Zervos (2007)). For a standard reference on irreversible investment, see Dixit and Pindyck (1994).…”