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.

On wald-type optimal stopping for Brownian motion

1997 ◽  
Vol 34 (01) ◽  
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 supτ E(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.

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.

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.

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.

Optimal stopping problems for Brownian motion

1970 ◽  
Vol 2 (2) ◽  
pp. 259-286 ◽  
Author(s):  
John Bather
Keyword(s):  
Brownian Motion ◽  
General Problem ◽  
Optimal Stopping ◽  
Stopping Time ◽  
Brownian Motion Process ◽  
Final Time ◽  
Optimal Stopping Time ◽  
Motion Process ◽  
Optimal Stopping Problems ◽  
The Cost

This paper is concerned with the general problem of choosing an optimal stopping time for a Brownian motion process, where the cost associated with any trajectory depends only on its final time and position.

Optimal stopping problems for Brownian motion

1970 ◽  
Vol 2 (02) ◽  
pp. 259-286 ◽  
Author(s):  
John Bather
Keyword(s):  
Brownian Motion ◽  
General Problem ◽  
Optimal Stopping ◽  
Stopping Time ◽  
Brownian Motion Process ◽  
Final Time ◽  
Optimal Stopping Time ◽  
Motion Process ◽  
Optimal Stopping Problems ◽  
The Cost

This paper is concerned with the general problem of choosing an optimal stopping time for a Brownian motion process, where the cost associated with any trajectory depends only on its final time and position.

Optimal stopping problems with generalized objective functions

1990 ◽  
Vol 27 (4) ◽  
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 of EXt, where t is 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.

Optimal Stopping of the Maximum Process

2014 ◽  
Vol 51 (03) ◽  
pp. 818-836 ◽  
Author(s):  
Luis H. R. Alvarez ◽  
Pekka Matomäki
Keyword(s):  
Brownian Motion ◽  
Free Boundary ◽  
Optimal Stopping ◽  
Original Problem ◽  
Alternative Route ◽  
Linear Diffusion ◽  
Maximum Process ◽  
Running Maximum ◽  
Parameterized Family ◽  
Optimal Stopping Problems

We consider a class of optimal stopping problems involving both the running maximum as well as the prevailing state of a linear diffusion. Instead of tackling the problem directly via the standard free boundary approach, we take an alternative route and present a parameterized family of standard stopping problems of the underlying diffusion. We apply this family to delineate circumstances under which the original problem admits a unique, well-defined solution. We then develop a discretized approach resulting in a numerical algorithm for solving the considered class of stopping problems. We illustrate the use of the algorithm in both a geometric Brownian motion and a mean reverting diffusion setting.

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