Optimal Control on the $L^\infty $ Norm of a Diffusion Process

Abstract

Stochastic control problems are considered, where the cost to be minimized is either a running maximum of the state variable or more generally a running maximum of a function of the state variable and the control. In both cases it is proved that the value function, which must be defined on an augmented state space to take care of the non-Markovian feature of the running maximum, is the unique viscosity solution of the associated Bellman equation, which turns out to be, in the second case, a variational inequality with an oblique derivative boundary condition. Most of this work consists of proving the convergence of L p approximations and this is done by purely partial differential equation (PDE) methods.

Correspondence between lifetime minimum wealth and utility of consumption

“…We state this formally and without proof in the following corollary, which is similar to a result of Barles, Daher, and Romano (1994).…”

Optimal lifetime consumption and investment under a drawdown constraint

“…The optimal consumption-investment problem (2.17) is in the class of stochastic control problems studied in Barles, Daher and Romano [1]. The dynamic programming equation is related to the second order operator…”

Dynamic Programming and Error Estimates for Stochastic Control Problems with Maximum Cost

“…A detailed description of this model can be found in [38]. Typical payoff functions are ψ(x, y) = y − x (lookback floating strike put), ψ(x, y) = max(y − E, 0) (fixed strike lookback call), ψ(x, y) = max(min(y, E) − x, 0) (lookback limited-risk put), etc., see [38,5] (see also [4] for other examples and related american lookback options).…”

“…Stochastic optimal control problems with running maximum cost in the viscosity solutions framework have been studied in [4,11]. The arguments developed in that papers are based on the approximation technique of the L ∞ -norm and they only apply if ψ and g are positive functions.…”

“…Another application of control problem with maximum running cost comes from the characterization of reachable sets for state-constrained controlled stochastic systems (see section 2.3). The main contributions to the study of this kind of problems can be found in [4] and [11] (see also [26] for the stationary case in the framework of classical solutions). In these works the dynamic programming techniques are applied on the L p -approximation of the L ∞ -cost functional (1.2), using the approximation:…”