|COOKIES: By using this website you agree that we can place Google Analytics Cookies on your device for performance monitoring.|
Unbiased Shifts for Brownian Motion
If you have a question about this talk, please contact Julia Blackwell.
Unbiased shifts of Brownian motion Based on joint work with Günter Last and Peter Mörters
Let B = (B(t) : t in R) be a two-sided standard Brownian motion. Let T be a real-valued measurable function of B. If T is a nonnegative stopping time then the shifted process (B(T + t) – B(T) : t nonnegative) is a one-sided Brownian motion independent of B(T). However, the two-sided process (B(T + t) – B(T) : t in R) need not be a Brownian motion. Moreover, the example of a fixed time T = s shows that even if it is, it need not be independent of B(T).
Call T an unbiased shift of B if (B(T + t) – B(T) : t in R) is a Brownian motion independent of B(T). Unbiased shifts can be characterized in terms of allocation rules balancing additive functionals of B. For any probability distribution Q on R we construct a stopping time T with the above properties such that B(T) has distribution Q. Also moment and minimality properties of unbiased shifts are given.
The case when Q is concentrated at zero is of special interest. We obtain a rigorous formulation of the intuitive idea that B looks globally the same from all its zeros, thus resolving an issue raised by Mandelbrot in The Fractal Geometry of Nature. The result can be stated as follows: if we travel in time according to the clock of local time we always see a two-sided Brownian motion.
This talk is part of the Probability series.
This talk is included in these lists:
Note that ex-directory lists are not shown.
Other listsThin-Film Magnetism Group The Triple Helix Lecture Series Molecular Techniques Seminars
Other talksProduction Processes Group Seminar Are Haemostasis and Thrombus Formation Regulated by Platelet Subpopulations? (Focus on Nitric Oxide Signalling) Risk prediction using common genetic variants Day 2 - Corporate Finance Theory Symposium 2015 The 2015 Vaccine Summit Indentation of a viscoelastic half-space