Maximal inequalities for reflected Brownian motion with drift. (English. Russian original) Zbl 0990.60078

Peshkir, G.; Shiryaev, A. N.

Theory Probab. Math. Stat. 63, 137-143 (2001); translation from Teor. Jmovirn. Mat. Stat. 63, 125-131 (2000).

Let \(F_{\mu}(x)=(e^{2\mu x}-2\mu x-1)/2\mu^2\) for \(x\geq 0\), and let \(H_{\mu}(x)=F^{-1}_{\mu}(x)\). The authors prove that if \(X=(X_{t})_{t\geq 0}\) is a reflected Brownian motion with drift \(-\mu, \mu>0\), and \(X_0=0\), then there exist constants \(c_1>0, c_2>0\) such that \[ c_1E(H_{\mu}(\tau))\leq E\Bigl(\max_{0\leq t\leq \tau}X_{t}\Bigr)\leq c_2E(H_{\mu}(\tau)) \] for all stopping times \(\tau\) for \(X\). Also it is proved that \[ E\Bigl(\max_{0\leq t\leq \tau}X_{t}\Bigr)\leq \inf_{c>0}(cE(\tau)+(\log(1+\mu/c))/2\mu) \] for all stopping times \(\tau\) for \(X\).

Reviewer: A.D.Borisenko (Kyïv)

MSC:

60J65	Brownian motion
60G40	Stopping times; optimal stopping problems; gambling theory

Keywords:

inequalities; maximum; reflected Brownian motion; drift; optimal stopping