Optimal stopping of strong Markov processes

Christensen, Sören; Salminen, Paavo; Ta, Bao Quoc

doi:10.1016/j.spa.2012.11.006

Cited by 36 publications

(48 citation statements)

References 23 publications

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“…As a more advanced application of the method in the previous example, we consider a Markov process on the real line. To construct an equilibrium stopping time for a wide class of examples, we consider the general setting of [12] for the auxiliary optimal stopping problems with value function v y (x) = sup τ E x (e −rτ F (X τ , y)), y ∈ R. (4.1)…”

Section: A Class Of One-sided Solvable Problems With Potential Jumpsmentioning

confidence: 99%

On Finding Equilibrium Stopping Times for Time-Inconsistent Markovian Problems

Christensen¹,

Lindensjö²

2018

SIAM J. Control Optim.

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Standard Markovian optimal stopping problems are consistent in the sense that the first entrance time into the stopping set is optimal for each initial state of the process. Clearly, the usual concept of optimality cannot in a straightforward way be applied to non-standard stopping problems without this time-consistent structure. This paper is devoted to the solution of time-inconsistent stopping problems with the reward depending on the initial state using an adaptation of Strotz's consistent planning. More precisely, we give a precise equilibrium definition -of the type subgame perfect Nash equilibrium based on pure Markov strategies. In general, such equilibria do not always exist and if they exist they are in general not unique. We, however, develop an iterative approach to finding equilibrium stopping times for a general class of problems and apply this approach to one-sided stopping problems on the real line. We furthermore prove a verification theorem based on a set of variational inequalities which also allows us to find equilibria. In the case of a standard optimal stopping problem, we investigate the connection between the notion of an optimal and an equilibrium stopping time. As an application of the developed theory we study a selling strategy problem under exponential utility and endogenous habit formation.

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Section: A Class Of One-sided Solvable Problems With Potential Jumpsmentioning

confidence: 99%

On Finding Equilibrium Stopping Times for Time-Inconsistent Markovian Problems

Christensen¹,

Lindensjö²

2018

SIAM J. Control Optim.

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“…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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“…The optimal threshold is then found to be the root of Q ν . Using the results discussed in Subsection 7.7 below, it follows from the discussion in Christensen et al (2013), Subsection 3.3, that Q ν coincides with our function f from (3). As the roots of f (y) =…”

Section: The Novikov-shiryaev Problemmentioning

confidence: 65%

“…The results for random walks (and Lévy processes) are based on the Wiener-Hopf factorization and the results in the general Markov case heavily rely on representation results for excessive functions. Another main difference is that in this paper we give an explicit formula for the function f , which allows to obtain general results such as Theorem 6.1, whereas the existence of a representing functionf in (7) is not clear in general (but see the discussion in Christensen et al (2013), Subsection 2.2 and -for the Lévy process case using Fourier techniques -Lemma 3.1 in Surya (2007)). Another related approach is described in Woodroofe et al (1994).…”

Section: Connection To Other Approachesmentioning

confidence: 99%

A general method for finding the optimal threshold in discrete time

Christensen

Irle

2018

Stochastics

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We develop an approach for solving one-sided optimal stopping problems in discrete time for general underlying Markov processes on the real line. The main idea is to transform the problem into an auxiliary problem for the ladder height variables. In case that the original problem has a one-sided solution and the auxiliary problem has a monotone structure, the corresponding myopic stopping time is optimal for the original problem as well. This elementary line of argument directly leads to a characterization of the optimal boundary in the original problem: The optimal threshold is given by the threshold of the myopic stopping time in the auxiliary problem. Supplying also a sufficient condition for our approach to work, we obtain solutions for many prominent examples in the literature, among others the problems of Novikov-Shiryaev, Shepp-Shiryaev, and the American put in option pricing under general conditions. As a further application we show that for underlying random walks (and Lévy processes in continuous time), general monotone and log-concave reward functions g lead to one-sided stopping problems.

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Optimal stopping of strong Markov processes

Cited by 36 publications

References 23 publications

On Finding Equilibrium Stopping Times for Time-Inconsistent Markovian Problems

On Finding Equilibrium Stopping Times for Time-Inconsistent Markovian Problems

Optimal stopping of Brownian motion with broken drift

A general method for finding the optimal threshold in discrete time

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