Optimal stopping of the maximum process: the maximality principle

Peškir, Goran

doi:10.1214/aop/1022855875

Cited by 98 publications

(115 citation statements)

References 18 publications

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“…Generalized drawdown times appear naturally in the Azema Yor solution of the Skorokhod embedding problem Azéma and Yor (1979), and in the Dubbins-Shepp-Shiryaev, and Peskir-Hobson-Egami optimal stopping problems Dubins et al (1994); Egami and Oryu (2015); Hobson (2007); Peskir (1998). Importantly, they allow a unified treatment of classic first passage and drawdown times (see also Avram et al (2018b) for a further generalization to taxed processes)-see Avram et al (2017b); Li et al (2017).…”

Section: Generalized Draw-down Stopping For Processes Without Positivmentioning

confidence: 99%

The W,Z/ν,δ Paradigm for the First Passage of Strong Markov Processes without Positive Jumps

Avram

Grahovac

Vardar-Acar

2019

Risks

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As is well-known, the benefit of restricting Lévy processes without positive jumps is the “ W , Z scale functions paradigm”, by which the knowledge of the scale functions W , Z extends immediately to other risk control problems. The same is true largely for strong Markov processes X t , with the notable distinctions that (a) it is more convenient to use as “basis” differential exit functions ν , δ , and that (b) it is not yet known how to compute ν , δ or W , Z beyond the Lévy, diffusion, and a few other cases. The unifying framework outlined in this paper suggests, however, via an example that the spectrally negative Markov and Lévy cases are very similar (except for the level of work involved in computing the basic functions ν , δ ). We illustrate the potential of the unified framework by introducing a new objective () for the optimization of dividends, inspired by the de Finetti problem of maximizing expected discounted cumulative dividends until ruin, where we replace ruin with an optimally chosen Azema-Yor/generalized draw-down/regret/trailing stopping time. This is defined as a hitting time of the “draw-down” process Y t = sup 0 ≤ s ≤ t X s - X t obtained by reflecting X t at its maximum. This new variational problem has been solved in a parallel paper.

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Section: Generalized Draw-down Stopping For Processes Without Positivmentioning

confidence: 99%

The W,Z/ν,δ Paradigm for the First Passage of Strong Markov Processes without Positive Jumps

Avram

Grahovac

Vardar-Acar

2019

Risks

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“…Motivated by facts from [11], consider the following optimal stopping problem: Then it is possible to verify that the map s 7 ! h 3 (s) given through its inverse by 1 which stays strictly below the diagonal in IR 2 , and thus by the maximality principle (see [10]) the stopping time 3 is optimal for the problem (3.49 which is treated easily in this context. We omit all remaining details for simplicity.…”

Section: Proposition 32 (The Minimax Property)mentioning

confidence: 99%

The Azéma-Yor Embedding in Brownian Motion with Drift

Peškir

2000

High Dimensional Probability II

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Let B = (B t ) t0 be standard Brownian motion started at zero, let > 0 be given and fixed, and let be a probability measure on IR having a strictly positive density F 0 . Then there exists a stopping time 3 of B such that In addition, it is proved that 3 is pointwise the smallest possible stopping time satisfying (B 3 + 3 ) which generates stochastically the largest possible maximum of the process (B t +t) t0 up to the time of stopping. This minimax property characterizes 3 uniquely. The result recovers the Azéma-Yor solution of the Skorokhod embedding problem [1] by passing to the limit when # 0 . The condition on the existence of a strictly positive density is imposed for simplicity, and more general cases can be treated similarly. The line of arguments used in the proof can be extended to treat the case of more general nonrecurrent diffusions.

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“…The problems of pricing of perpetual American lookback and other options with more complicated structure of payoffs depending on the running maxima of the underlying processes were studied in Baurdoux andKyprianou (2009), Gapeev (2007), Guo and Zervos (2010), Ott (2013), Kyprianou and Ott (2014), and Rodosthenous and Zervos (2017) among others (see also Gapeev (2006) and Kitapbayev (2014) for the finite time horizon American lookback options on the running maxima of geometric Brownian motions). Along with the article of Dubins, Shepp, and Shiryaev (1993), the papers of Shiryaev (1993, 1994) also made a crucial contribution to the optimal stopping problems arising in the proofs of maximal inequalities for the continuous time processes further developed in Graversen and Peskir (1998a,b) and Peskir (1998) among others (see also Peskir and Shiryaev (2006); Chapter V for an extensive overview of the optimal stopping problems related to maximal inequalities). Shepp and Shiryaev (1996) proposed and explicitly solved the dual Russian (call) option pricing problem of (2.4) as an optimal stopping problem for a two-dimensional continuous Markov process (X,Y) = (X t ,Y t ) t≥0 defined in (2.1-2.3).…”

Section: Introductionmentioning

confidence: 99%

Solving the dual Russian option problem by using change‐of‐measure arguments

Gapeev

2019

High Frequency

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We apply the change‐of‐measure arguments of Shepp and Shiryaev (Theory of Probability and its Applications, 1994, 39, 103–119) to study the dual Russian option pricing problem proposed by Shepp and Shiryaev (Probability Theory and Mathematical Statistics: Lectures presented at the semester held in St. Peterburg, Russia, March 2 April 23, 1993, Amsterdam, the Netherlands: Gordon and Breach, 1996, pp. 209–218) as an optimal stopping problem for a one‐dimensional diffusion process with reflection. We recall the solution to the associated free‐boundary problem and give a solution to the resulting one‐dimensional optimal stopping problem by using the martingale approach of Beibel and Lerche (Statistica Sinica, 1997, 7, 93–108) and (Theory of Probability and its Applications, 2000, 45, 657–669).

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Optimal stopping of the maximum process: the maximality principle

Cited by 98 publications

References 18 publications

The W,Z/ν,δ Paradigm for the First Passage of Strong Markov Processes without Positive Jumps

The W,Z/ν,δ Paradigm for the First Passage of Strong Markov Processes without Positive Jumps

The Azéma-Yor Embedding in Brownian Motion with Drift

Solving the dual Russian option problem by using change‐of‐measure arguments

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