On the threshold strategies in optimal stopping problems for diffusion processes

Аркин, В. И.; Slastnikov, Alexander

doi:10.1017/jpr.2017.44

Cited by 3 publications

(3 citation statements)

References 7 publications

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“…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

τ^{*} = inf false{t \geq 0 0pt: X (t) \notin (x_{*}, x^{*}) false} .

…”

Section: Introductionmentioning

confidence: 99%

Optimal stopping of Brownian motion with broken drift

Mordecki¹,

Salminen

2019

High Frequency

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We solve an optimal stopping problem where the underlying diffusion is Brownian motion on boldR with a positive piecewise constant drift changing at zero. It is assumed that the drift μ1 on the negative side is smaller than the drift μ2 on the positive side. The main observation is that if μ2−μ1>1/2, then there exist values of the discounting parameter for which it is not optimal to stop in the vicinity of zero where the drift changes. However, when the discounting gets larger, the stopping region becomes connected and contains zero. This is in contrast with results concerning optimal stopping of skew Brownian motion where the skew point is for all values of the discounting parameter in the continuation region in case the skew parameter is bigger than 1/2. In all cases, a practical implementation of these optimal stopping strategies, based on hitting times of boundaries of continuation regions, would require access to the highest‐possible frequency data which is consistent with the continuous‐time modeling framework.

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τ^{*} = inf false{t \geq 0 0pt: X (t) \notin (x_{*}, x^{*}) false} .

…”

Section: Introductionmentioning

confidence: 99%

Optimal stopping of Brownian motion with broken drift

Mordecki¹,

Salminen

2019

High Frequency

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“…For the optimal stopping problem under the Lévy processes, refer to Mordecki & Mishura (2016), Lin & Yao (2019), and Mordecki and Eguren (2021). For the optimal stopping problem under diffusion processes, see Lempa (2010), Lamberton et al (2013), and Arkin and Slastnikov (2017).…”

Section: Introductionmentioning

confidence: 99%

“…For the optimal stopping problem under diffusion processes, see Lempa (2010), Lamberton et al. (2013), and Arkin and Slastnikov (2017).…”

Section: Introductionmentioning

confidence: 99%

A general approximation method for optimal stopping and random delay

Chen

Song

2023

Mathematical Finance

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This study examines the continuous‐time optimal stopping problem with an infinite horizon under Markov processes. Existing research focuses on finding explicit solutions under certain assumptions of the reward function or underlying process; however, these assumptions may either not be fulfilled or be difficult to validate in practice. We developed a continuous‐time Markov chain (CTMC) approximation method to find the optimal solution, which applies to general reward functions and underlying Markov processes. We demonstrated that our method can be used to solve the optimal stopping problem with a random delay, in which the delay could be either an independent random variable or a function of the underlying process. We established a theoretical upper bound for the approximation error to facilitate error control. Furthermore, we designed a two‐stage scheme to implement our method efficiently. The numerical results show that the proposed method is accurate and rapid under various model specifications.

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