Threshold Strategies in Optimal Stopping Problem for One-Dimensional Diffusion Processes

Abstract

We study a problem when a solution to optimal stopping problem for one-dimensional diffusion will generate by threshold strategy. Namely, we give necessary and sufficient conditions under which an optimal stopping time can be specified as the first time when the process exceeds some level (threshold), and a continuation set is a semi-interval. We give also second-order conditions, which allow to discard such solutions to free-boundary problem that are not the solutions to optimal stopping problem.

Optimal stopping of Brownian motion with broken drift

“…For typical examples of one‐sided solutions derived using the principle of smooth fit, we refer to papers on Russian options due to Shepp and Shiryaev (, ). Villeneuve () gives sufficient conditions to have threshold optimal strategies, and Arkin () gives necessary and sufficient conditions for Itô diffusions with C 2 payoffs functions to have one‐sided solutions, whereas Arkin and Slastnikov () and Crocce and Mordecki () give also necessary and sufficient conditions in different and more general diffusion frameworks. For more general Markov processes, Christensen and Irle (), Christensen, Salminen, and Ta (), and Mordecki and Salminen () propose verification results for one‐sided solutions, but also for problems where the optimal stopping time is of the form …”

Real Options and Threshold Strategies

“…ii) Now, let τ p * be optimal stopping time in the problem (2). Note, that τ p * will be an optimal stopping time in the problem (7) also.…”

“…Since this problem is a special case of optimal stopping problem, the similar question may be addressed to a general optimal stopping problem: Under what conditions (on both process and payoff function) an optimal stopping time will have a threshold structure? Some results in this direction (in the form of necessary and sufficient conditions) were obtained in [3], [5], [2] under some additional assumptions on underlying process and/or payoffs.…”

On the threshold strategies in optimal stopping problems for diffusion processes

“…where g : D → R 1 is the payoff function, ρ > 0 is the discount rate, 1 A is the indicator function of the set A, E x means the expectation for the process X t starting from the initial state x, and the supremum in (1) takes values over all stopping times τ (with respect to the natural filtration F t = σ {X s , 0 ≤ s ≤ t}, t ≥ 0). We consider stopping times which can take infinite values (with positive probability).…”

“…S = {x ∈ D : x ≥ x * } for some x * ; see [7,Theorem 4.2]. Later, Arkin [1] gave the necessary and sufficient conditions for the threshold structure of the stopping set in the case of Itô diffusion processes and the payoff functions from C 2 (D). Those conditions can be divided in three parts: to the left from threshold x * , to the right from x * , and locally at x * .…”