scholarly journals Optimal stopping problems with restricted stopping times

2017 ◽  
Vol 13 (1) ◽  
pp. 399-411 ◽  
Author(s):  
Wenqing Bao ◽  
◽  
Xianyi Wu ◽  
Xian Zhou ◽  
◽  
...  
Keyword(s):  
Optimal Stopping ◽  
Stopping Times ◽  
Optimal Stopping Problems

On wald-type optimal stopping for Brownian motion

1997 ◽  
Vol 34 (1) ◽  
pp. 66-73 ◽  
Author(s):  
S. E. Graversen ◽  
G. Peškir
Keyword(s):  
Brownian Motion ◽  
Optimal Stopping ◽  
Stopping Time ◽  
Maximal Inequality ◽  
Stopping Times ◽  
Standard Brownian Motion ◽  
Square Root ◽  
Real Analysis ◽  
Optimal Stopping Problems ◽  
Wald's Identity

The solution is presented to all optimal stopping problems of the form supτE(G(|Β τ |) – cτ), where is standard Brownian motion and the supremum is taken over all stopping times τ for B with finite expectation, while the map G : ℝ+ → ℝ satisfies for some being given and fixed. The optimal stopping time is shown to be the hitting time by the reflecting Brownian motion of the set of all (approximate) maximum points of the map . The method of proof relies upon Wald's identity for Brownian motion and simple real analysis arguments. A simple proof of the Dubins–Jacka–Schwarz–Shepp–Shiryaev (square root of two) maximal inequality for randomly stopped Brownian motion is given as an application.

scholarly journals On Asymptotics of Optimal Stopping Times

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 194
Author(s):  
Hugh N. Entwistle ◽  
Christopher J. Lustri ◽  
Georgy Yu. Sofronov
Keyword(s):  
Asymptotic Behaviour ◽  
Optimal Stopping ◽  
Upper Bound ◽  
Stopping Time ◽  
Stopping Times ◽  
Independent Random Variables ◽  
Algebraic Form ◽  
Asymptotic Expressions ◽  
Optimal Stopping Problems ◽  
Optimal Stopping Times

We consider optimal stopping problems, in which a sequence of independent random variables is drawn from a known continuous density. The objective of such problems is to find a procedure which maximizes the expected reward. In this analysis, we obtained asymptotic expressions for the expectation and variance of the optimal stopping time as the number of drawn variables became large. In the case of distributions with infinite upper bound, the asymptotic behaviour of these statistics depends solely on the algebraic power of the probability distribution decay rate in the upper limit. In the case of densities with finite upper bound, the asymptotic behaviour of these statistics depends on the algebraic form of the distribution near the finite upper bound. Explicit calculations are provided for several common probability density functions.

scholarly journals Discrete-type approximations for non-Markovian optimal stopping problems: Part I

2019 ◽  
Vol 56 (4) ◽  
pp. 981-1005 ◽  
Author(s):  
Dorival Leão ◽  
Alberto Ohashi ◽  
Francesco Russo
Keyword(s):  
Brownian Motion ◽  
Optimal Stopping ◽  
Stopping Time ◽  
Rates Of Convergence ◽  
Stopping Times ◽  
Full Generality ◽  
Optimal Values ◽  
Optimal Stopping Problems ◽  
Optimal Stopping Times ◽  
Discrete Type

AbstractWe present a discrete-type approximation scheme to solve continuous-time optimal stopping problems based on fully non-Markovian continuous processes adapted to the Brownian motion filtration. The approximations satisfy suitable variational inequalities which allow us to construct $\varepsilon$ -optimal stopping times and optimal values in full generality. Explicit rates of convergence are presented for optimal values based on reward functionals of path-dependent stochastic differential equations driven by fractional Brownian motion. In particular, the methodology allows us to design concrete Monte Carlo schemes for non-Markovian optimal stopping time problems as demonstrated in the companion paper by Bezerra et al.

Monotone stopping games

1987 ◽  
Vol 24 (02) ◽  
pp. 386-401 ◽  
Author(s):  
John W. Mamer
Keyword(s):  
Nash Equilibrium ◽  
Optimal Stopping ◽  
Nash Equilibria ◽  
Stopping Times ◽  
Optimal Stopping Problem ◽  
Optimal Stopping Problems ◽  
Zero Sum ◽  
Stopping Games

We consider the extension of optimal stopping problems to non-zero-sum strategic settings called stopping games. By imposing a monotone structure on the pay-offs of the game we establish the existence of a Nash equilibrium in non-randomized stopping times. As a corollary, we identify a class of games for which there are Nash equilibria in myopic stopping times. These games satisfy the strategic equivalent of the classical ‘monotone case' assumptions of the optimal stopping problem.

scholarly journals Optimal Stopping Problems in Diffusion-Type Models with Running Maxima and Drawdowns

2014 ◽  
Vol 51 (03) ◽  
pp. 799-817 ◽  
Author(s):  
Pavel V. Gapeev ◽  
Neofytos Rodosthenous
Keyword(s):  
Optimal Stopping ◽  
Free Boundary Problems ◽  
Three Dimensional ◽  
Stopping Times ◽  
Value Functions ◽  
Diffusion Type ◽  
Closed Form Solutions ◽  
Optimal Stopping Problems ◽  
Optimal Stopping Times ◽  
Maximum Drawdown

We study optimal stopping problems related to the pricing of perpetual American options in an extension of the Black-Merton-Scholes model in which the dividend and volatility rates of the underlying risky asset depend on the running values of its maximum and maximum drawdown. The optimal stopping times of the exercise are shown to be the first times at which the price of the underlying asset exits some regions restricted by certain boundaries depending on the running values of the associated maximum and maximum drawdown processes. We obtain closed-form solutions to the equivalent free-boundary problems for the value functions with smooth fit at the optimal stopping boundaries and normal reflection at the edges of the state space of the resulting three-dimensional Markov process. We derive first-order nonlinear ordinary differential equations for the optimal exercise boundaries of the perpetual American standard options.

scholarly journals Optimal Stopping Problems in Diffusion-Type Models with Running Maxima and Drawdowns

2014 ◽  
Vol 51 (3) ◽  
pp. 799-817 ◽  
Author(s):  
Pavel V. Gapeev ◽  
Neofytos Rodosthenous
Keyword(s):  
Optimal Stopping ◽  
Free Boundary Problems ◽  
Three Dimensional ◽  
Stopping Times ◽  
Value Functions ◽  
Diffusion Type ◽  
Closed Form Solutions ◽  
Optimal Stopping Problems ◽  
Optimal Stopping Times ◽  
Maximum Drawdown

We study optimal stopping problems related to the pricing of perpetual American options in an extension of the Black-Merton-Scholes model in which the dividend and volatility rates of the underlying risky asset depend on the running values of its maximum and maximum drawdown. The optimal stopping times of the exercise are shown to be the first times at which the price of the underlying asset exits some regions restricted by certain boundaries depending on the running values of the associated maximum and maximum drawdown processes. We obtain closed-form solutions to the equivalent free-boundary problems for the value functions with smooth fit at the optimal stopping boundaries and normal reflection at the edges of the state space of the resulting three-dimensional Markov process. We derive first-order nonlinear ordinary differential equations for the optimal exercise boundaries of the perpetual American standard options.

Optimal stopping problems with generalized objective functions

1990 ◽  
Vol 27 (04) ◽  
pp. 828-838
Author(s):  
T. P. Hill ◽  
D. P. Kennedy
Keyword(s):  
Optimal Stopping ◽  
Stopping Time ◽  
Random Variables ◽  
Stopping Times ◽  
Objective Functions ◽  
Monotone Functions ◽  
Optimal Stopping Problems ◽  
Universal Bounds ◽  
Optimal Stopping Times ◽  
Entire Vector

Optimal stopping of a sequence of random variables is studied, with emphasis on generalized objectives which may be non-monotone functions ofEXt, wheretis a stopping time, or may even depend on the entire vector (E[X1I{t=l}], · ··,E[XnI{t=n}]),such as the minimax objective to maximize minj{E[XjI{t=j}]}.Convexity is used to establish a prophet inequality and universal bounds for the optimal return, and a method for constructing optimal stopping times for such objectives is given.

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 Perpetual American Cancellable Standard Options in Models with Last Passage Times

Algorithms ◽  
2020 ◽  
Vol 14 (1) ◽  
pp. 3
Author(s):  
Pavel V. Gapeev ◽  
Libo Li ◽  
Zhuoshu Wu
Keyword(s):  
Optimal Stopping ◽  
Free Boundary Problems ◽  
Asset Price ◽  
Stopping Times ◽  
Hitting Times ◽  
Explicit Solutions ◽  
Boundary Problems ◽  
Price Process ◽  
Optimal Stopping Problems ◽  
Passage Times

We derive explicit solutions to the perpetual American cancellable standard put and call options in an extension of the Black–Merton–Scholes model. It is assumed that the contracts are cancelled at the last hitting times for the underlying asset price process of some constant upper or lower levels which are not stopping times with respect to the observable filtration. We show that the optimal exercise times are the first times at which the asset price reaches some lower or upper constant levels. The proof is based on the reduction of the original optimal stopping problems to the associated free-boundary problems and the solution of the latter problems by means of the smooth-fit conditions.

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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