1 results
On wald-type optimal stopping for Brownian motion
- Part of
-
- Journal:
- Journal of Applied Probability / Volume 34 / Issue 1 / March 1997
- Published online by Cambridge University Press:
- 14 July 2016, pp. 66-73
- Print publication:
- March 1997
-
- Article
- Export citation
-
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.
is standard Brownian motion and the supremum is taken over all stopping times 
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.