Randomised rules for stopping problems

Abstract

In a classical, continuous-time, optimal stopping problem, the agent chooses the best time to stop a stochastic process in order to maximise the expected discounted return. The agent can choose when to stop, and if at any moment they decide to stop, stopping occurs immediately with probability one. However, in many settings this is an idealistic oversimplification. Following Strack and Viefers we consider a modification of the problem in which stopping occurs at a rate which depends on the relative values of stopping and continuing: there are several different solutions depending on how the value of continuing is calculated. Initially we consider the case where stopping opportunities are constrained to be event times of an independent Poisson process. Motivated by the limiting case as the rate of the Poisson process increases to infinity, we also propose a continuous-time formulation of the problem where stopping can occur at any instant.

Solving penalised American options for jump diffusions using the POST algorithm

“…If x (0) = 0 then (18) generates a non-decreasing sequence {x (n) }, i.e. x (n) ≥ x (n−1) , such that {(x (n) , w (n) )} converges to a solution of (17).…”

“…In financial terms, this can be seen as imposing a "liquidity constraint" on the available exercise times. Other papers using Poisson-generated stopping times, though not the POST algorithm, include, in chronological order, [32,11,8,24,25,27,16,17]. In fact, the POST algorithm is used in [15, eq.…”