scholarly journals General drawdown of general tax model in a time-homogeneous Markov framework

2021 ◽  
Vol 58 (4) ◽  
pp. 1131-1151
Author(s):  
Florin Avram ◽  
Bin Li ◽  
Shu Li
Keyword(s):  
Markov Process ◽  
Optimal Stopping ◽  
Mathematical Finance ◽  
Levy Processes ◽  
Free Time ◽  
First Passage ◽  
Homogeneous Markov Process ◽  
Cusum Procedure ◽  
Optimal Stopping Problems ◽  
Spectrally Negative Lévy Processes

AbstractDrawdown/regret times feature prominently in optimal stopping problems, in statistics (CUSUM procedure), and in mathematical finance (Russian options). Recently it was discovered that a first passage theory with more general drawdown times, which generalize classic ruin times, may be explicitly developed for spectrally negative Lévy processes [9, 20]. In this paper we further examine the general drawdown-related quantities in the (upward skip-free) time-homogeneous Markov process, and then in its (general) tax process by noticing the pathwise connection between general drawdown and the tax process.

On Optimal Stopping of Inhomogeneous Standard Markov Processes

1995 ◽  
Vol 2 (4) ◽  
pp. 335-346
Author(s):  
B. Dochviri
Keyword(s):  
Markov Process ◽  
Optimal Stopping ◽  
Value Function ◽  
Stopping Times ◽  
Homogeneous Markov Process ◽  
Limit Procedure ◽  
Optimal Stopping Problems ◽  
Optimal Stopping Times ◽  
The Value Function

Abstract The connection between the optimal stopping problems for inhomogeneous standard Markov process and the corresponding homogeneous Markov process constructed in the extended state space is established. An excessive characterization of the value-function and the limit procedure for its construction in the problem of optimal stopping of an inhomogeneous standard Markov process is given. The form of ε-optimal (optimal) stopping times is also found.

scholarly journals The W, Z scale functions kit for first passage problems of spectrally negative Lévy processes, and applications to control problems

2020 ◽  
Vol 24 ◽  
pp. 454-525 ◽  
Author(s):  
Florin Avram ◽  
Danijel Grahovac ◽  
Ceren Vardar-Acar
Keyword(s):  
Lévy Processes ◽  
Mathematical Finance ◽  
Levy Processes ◽  
Small Sample ◽  
First Passage ◽  
Scale Functions ◽  
Exit Problems ◽  
Finance Risk ◽  
Storage Theory ◽  
Spectrally Negative Lévy Processes

In the last years there appeared a great variety of identities for first passage problems of spectrally negative Lévy processes, which can all be expressed in terms of two “q-harmonic functions” (or scale functions) W and Z. The reason behind that is that there are two ways of exiting an interval, and thus two fundamental “two-sided exit” problems from an interval (TSE). Since many other problems can be reduced to TSE, researchers developed in the last years a kit of formulas expressed in terms of the “W, Z alphabet”. It is important to note – as is currently being shown – that these identities apply equally to other spectrally negative Markov processes, where however the W, Z functions are typically much harder to compute. We collect below our favorite recipes from the “W, Z kit”, drawing from various applications in mathematical finance, risk, queueing, and inventory/storage theory. A small sample of applications concerning extensions of the classic de Finetti dividend problem is offered. An interesting use of the kit is for recognizing relationships between problems involving behaviors apparently unrelated at first sight (like reflection, absorption, etc.). Another is expressing results in a standardized form, improving thus the possibility to check when a formula is already known.

scholarly journals On the Continuous and Smooth Fit Principle for Optimal Stopping Problems in Spectrally Negative Lévy Models

2014 ◽  
Vol 46 (1) ◽  
pp. 139-167 ◽  
Author(s):  
Masahiko Egami ◽  
Kazutoshi Yamazaki
Keyword(s):  
Optimal Stopping ◽  
Optimal Stopping Problem ◽  
Infinite Time ◽  
Expected Payoff ◽  
Threshold Levels ◽  
Put Option ◽  
Optimal Stopping Problems ◽  
American Put ◽  
Spectrally Negative Lévy Processes ◽  
Optimal Candidate

We consider a class of infinite time horizon optimal stopping problems for spectrally negative Lévy processes. Focusing on strategies of threshold type, we write explicit expressions for the corresponding expected payoff via the scale function, and further pursue optimal candidate threshold levels. We obtain and show the equivalence of the continuous/smooth fit condition and the first-order condition for maximization over threshold levels. As examples of its applications, we give a short proof of the McKean optimal stopping problem (perpetual American put option) and solve an extension to Egami and Yamazaki (2013).

scholarly journals On the Continuous and Smooth Fit Principle for Optimal Stopping Problems in Spectrally Negative Lévy Models

2014 ◽  
Vol 46 (01) ◽  
pp. 139-167 ◽  
Author(s):  
Masahiko Egami ◽  
Kazutoshi Yamazaki
Keyword(s):  
Optimal Stopping ◽  
Optimal Stopping Problem ◽  
Infinite Time ◽  
Expected Payoff ◽  
Threshold Levels ◽  
Put Option ◽  
Optimal Stopping Problems ◽  
American Put ◽  
Spectrally Negative Lévy Processes ◽  
Optimal Candidate

We consider a class of infinite time horizon optimal stopping problems for spectrally negative Lévy processes. Focusing on strategies of threshold type, we write explicit expressions for the corresponding expected payoff via the scale function, and further pursue optimal candidate threshold levels. We obtain and show the equivalence of the continuous/smooth fit condition and the first-order condition for maximization over threshold levels. As examples of its applications, we give a short proof of the McKean optimal stopping problem (perpetual American put option) and solve an extension to Egami and Yamazaki (2013).

scholarly journals One-sided solutions for optimal stopping problems with logconcave reward functions

2019 ◽  
Vol 51 (01) ◽  
pp. 87-115
Author(s):  
Yi-Shen Lin ◽  
Yi-Ching Yao
Keyword(s):  
Random Walks ◽  
Optimal Stopping ◽  
Lévy Processes ◽  
Levy Processes ◽  
Stopping Times ◽  
Reward Function ◽  
Continuous Reward ◽  
Optimal Stopping Problems ◽  
Reward Functions ◽  
Optimal Stopping Times

AbstractIn the literature on optimal stopping, the problem of maximizing the expected discounted reward over all stopping times has been explicitly solved for some special reward functions (including (x+)ν, (ex − K)+, (K − e− x)+, x ∈ ℝ, ν ∈ (0, ∞), and K > 0) under general random walks in discrete time and Lévy processes in continuous time (subject to mild integrability conditions). All such reward functions are continuous, increasing, and logconcave while the corresponding optimal stopping times are of threshold type (i.e. the solutions are one-sided). In this paper we show that all optimal stopping problems with increasing, logconcave, and right-continuous reward functions admit one-sided solutions for general random walks and Lévy processes, thereby generalizing the aforementioned results. We also investigate in detail the principle of smooth fit for Lévy processes when the reward function is increasing and logconcave.

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