Constructing time-homogeneous generalized diffusions consistent with optimal stopping values

Abstract

Consider a set of discounted optimal stopping problems for a one-parameter family of objective functions and a fixed diffusion process, started at a fixed point. A standard problem in stochastic control/optimal stopping is to solve for the problem value in this setting.In this article we consider an inverse problem; given the set of problem values for a family of objective functions, we aim to recover the diffusion. Under a natural assumption on the family of objective functions we can characterise existence and uniqueness of a diffusion for which the optimal stopping problems have the specified values. The solution of the problem relies on techniques from generalised convexity theory.

An explicit martingale version of the one-dimensional Brenier’s Theorem with full marginals constraint

“…This methodology generated developments in many directions, namely for different derivative contracts and/or multiple-marginals constraints, see e.g. Brown, Hobson & Rogers [13], Madan & Yor [62], Cox, Hobson & Oblój [18], Cox & Oblój [19,20], Davis, Oblój & Raval [23], Cox & Wang [22], Gassiat, Oberhauser & dos Reis [34], Cox, Oblój & Touzi [21], Hobson & Neuberger [51], and Hobson & Klimmek [47,48,49,50]. We also refer to the survey papers by Oblój [63] and Hobson [45] for more details.…”

The Wronskian parametrises the class of diffusions with a given distribution at a random time

“…Finally, this article is related to the inverse problem of constructing diffusions consistent with prices for perpetual American options or, more generally, with given value functions for perpetual horizon stopping problems, see Hobson and Klimmek [6]. As in this article, the underlying key idea in [6] is to construct consistent diffusions through the speed measure via the eigenfunctions of the diffusion.…”

Parameter Dependent Optimal Thresholds, Indifference Levels and Inverse Optimal Stopping Problems

“…As in Hobson and Klimmek [17], we will use generalized convex analysis of the log-transformed stopping problem to solve the inverse problem.…”

“…As an analysis of allocation indices and stopping problems, this article can be seen to extend the work of Karatzas [19]. However, the aim here is not to prove the optimality of the 'play-the-leader' policy for multi-armed bandits, but to generalise the approach to inverse optimal stopping problems introduced in [17]. The fundamental aim is to establish qualitative principles that govern the relationship between data (e.g.…”

“…In related work by Lu [21], the approach is developed to establish duality when the parameter space is a discrete set of strikes. More generally, Hobson and Klimmek [17] employ generalized convex analysis to establish duality between log-transformed value functions and log-transformed diffusion eigenfunctions for a general class of reward functions. The common strand in this previous work on inverse stopping problems is the conversion of a stochastic problem into a deterministic duality relation involving monotone optimizers.…”