scholarly journals Stopping times on Brownian motion: Some properties of root's construction

1970 ◽  
Vol 16 (3) ◽  
pp. 211-218 ◽  
Author(s):  
R. M. Loynes
Keyword(s):  
Brownian Motion ◽  
Stopping Times

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.

Optimal stopping and maximal inequalities for geometric Brownian motion

1998 ◽  
Vol 35 (04) ◽  
pp. 856-872 ◽  
Author(s):  
S. E. Graversen ◽  
G. Peskir
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Optimal Stopping ◽  
Moving Boundary ◽  
Stopping Time ◽  
Practical Interest ◽  
Maximal Inequality ◽  
Geometric Brownian Motion ◽  
Stopping Times ◽  
Image Position

Explicit formulas are found for the payoff and the optimal stopping strategy of the optimal stopping problem supτ E (max0≤t≤τ X t − c τ), where X = (X t ) t≥0 is geometric Brownian motion with drift μ and volatility σ > 0, and the supremum is taken over all stopping times for X. The payoff is shown to be finite, if and only if μ < 0. The optimal stopping time is given by τ* = inf {t > 0 | X t = g * (max0≤t≤s X s )} where s ↦ g *(s) is the maximal solution of the (nonlinear) differential equation under the condition 0 < g(s) < s, where Δ = 1 − 2μ / σ2 and K = Δ σ2 / 2c. The estimate is established g *(s) ∼ ((Δ − 1) / K Δ)1 / Δ s 1−1/Δ as s → ∞. Applying these results we prove the following maximal inequality: where τ may be any stopping time for X. This extends the well-known identity E (sup t>0 X t ) = 1 − (σ 2 / 2 μ) and is shown to be sharp. The method of proof relies upon a smooth pasting guess (for the Stephan problem with moving boundary) and the Itô–Tanaka formula (being applied two-dimensionally). The key point and main novelty in our approach is the maximality principle for the moving boundary (the optimal stopping boundary is the maximal solution of the differential equation obtained by a smooth pasting guess). We think that this principle is by itself of theoretical and practical interest.

Optimal stopping and maximal inequalities for geometric Brownian motion

1998 ◽  
Vol 35 (4) ◽  
pp. 856-872 ◽  
Author(s):  
S. E. Graversen ◽  
G. Peskir
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Optimal Stopping ◽  
Moving Boundary ◽  
Stopping Time ◽  
Practical Interest ◽  
Maximal Inequality ◽  
Geometric Brownian Motion ◽  
Stopping Times ◽  
Image Position

Explicit formulas are found for the payoff and the optimal stopping strategy of the optimal stopping problem supτE (max0≤t≤τXt − c τ), where X = (Xt)t≥0 is geometric Brownian motion with drift μ and volatility σ > 0, and the supremum is taken over all stopping times for X. The payoff is shown to be finite, if and only if μ < 0. The optimal stopping time is given by τ* = inf {t > 0 | Xt = g* (max0≤t≤sXs)} where s ↦ g*(s) is the maximal solution of the (nonlinear) differential equation under the condition 0 < g(s) < s, where Δ = 1 − 2μ / σ2 and K = Δ σ2 / 2c. The estimate is established g*(s) ∼ ((Δ − 1) / K Δ)1 / Δs1−1/Δ as s → ∞. Applying these results we prove the following maximal inequality: where τ may be any stopping time for X. This extends the well-known identity E (supt>0Xt) = 1 − (σ 2 / 2 μ) and is shown to be sharp. The method of proof relies upon a smooth pasting guess (for the Stephan problem with moving boundary) and the Itô–Tanaka formula (being applied two-dimensionally). The key point and main novelty in our approach is the maximality principle for the moving boundary (the optimal stopping boundary is the maximal solution of the differential equation obtained by a smooth pasting guess). We think that this principle is by itself of theoretical and practical interest.

SKOROKHOD EMBEDDINGS IN BOUNDED TIME

2011 ◽  
Vol 11 (02n03) ◽  
pp. 215-226 ◽  
Author(s):  
STEFAN ANKIRCHNER ◽  
PHILIPP STRACK
Keyword(s):  
Brownian Motion ◽  
Time Horizon ◽  
Sufficient Conditions ◽  
Stopping Time ◽  
Fixed Time ◽  
Stopping Times ◽  
Embedding Problem ◽  
Brownian Motion With Drift ◽  
Necessary And Sufficient ◽  
Skorokhod Embedding Problem

This article deals with the Skorokhod embedding problem in bounded time for the Brownian motion with drift Xt = κt + Wt: Given a probability measure μ we aim at finding a stopping time τ such that the law of Xτ is μ, and τ is almost surely smaller than some given fixed time horizon T > 0. We provide necessary and sufficient conditions on the distribution μ for the existence of such bounded stopping times.

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.

Excursions height- and length-related stopping times, and application to finance

2002 ◽  
Vol 34 (4) ◽  
pp. 846-868 ◽  
Author(s):  
Laurent Gauthier
Keyword(s):  
Brownian Motion ◽  
Laplace Transform ◽  
Stopping Time ◽  
Stopping Times ◽  
Excursion Theory ◽  
Delay Constraint ◽  
Investment Projects ◽  
The Laplace Transform

In this paper, we study the first instant when Brownian motion either spends consecutively more than a certain time above a certain level, or reaches another level. This stopping time generalizes the ‘Parisian’ stopping times that were introduced by Chesneyet al.(1997). Using excursion theory, we derive the Laplace transform of this stopping time. We apply this result to the valuation of investment projects with a delay constraint, but with an alternative: pay a higher cost and get the project started immediately

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