Controlling the Velocity of Brownian Motion by its Terminal Value

Let V = (Vt)t 0 be the Ornstein-Uhlenbeck velocity process solvin

Maximal inequalities for bessel processes

<p/> <p>It is proved that the uniform law of large numbers (over a random parameter set) for the <inline-formula><graphic file="1029-242X-1998-621735-i1.gif"/></inline-formula>-dimensional ( <inline-formula><graphic file="1029-242X-1998-621735-i2.gif"/></inline-formula>) Bessel process <inline-formula><graphic file="1029-242X-1998-621735-i3.gif"/></inline-formula> started at 0 is valid: <inline-formula><graphic file="1029-242X-1998-621735-i4.gif"/></inline-formula> for all stopping times <inline-formula><graphic file="1029-242X-1998-621735-i5.gif"/></inline-formula> for <inline-formula><graphic file="1029-242X-1998-621735-i6.gif"/></inline-formula>. The rate obtained (on the right-hand side) is shown to be the best possible. The following inequality is gained as a consequence: <inline-formula><graphic file="1029-242X-1998-621735-i7.gif"/></inline-formula> for all stopping times <inline-formula><graphic file="1029-242X-1998-621735-i8.gif"/></inline-formula> for <inline-formula><graphic file="1029-242X-1998-621735-i9.gif"/></inline-formula>, where the constant <inline-formula><graphic file="1029-242X-1998-621735-i10.gif"/></inline-formula> satisfies <inline-formula><graphic file="1029-242X-1998-621735-i11.gif"/></inline-formula> as <inline-formula><graphic file="1029-242X-1998-621735-i12.gif"/></inline-formula>. This answers a question raised in [4]. The method of proof relies upon representing the Bessel process as a time changed geometric Brownian motion. The main emphasis of the paper is on the method of proof and on the simplicity of solution.</p

Innovative HEaring Aid Research – Ecological Conditions and Outcome measures: the HEAR-ECO project

Six research studies for creating outcome measures that are more ecologically valid when assessing hearing aid performance