We use recent results on the Fourier analysis of the zero sets of Brownian motion to explore the diophantine properties of an algorithmically random Brownian motion ( also known as a complex oscillation). We discuss the construction and definability of perfect sets which are linearly independent over the rationals directly from Martin-Löf random reals. Finally we explore the recent work of Tsir...

2 Principles 4 2.1 Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Definition and history . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.2 Fractal dimension . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.3 Hölder exponent and singularity spectrum . . . . . . . . . . . . . 6 2.2 Self-similar random processes . . . . . . . . . . . . . . ....

Abstract We study optimal stopping problems for some functionals of Brownian motion in the case when the decision whether or not to stop before (or at) time t is allowed to be based on the δ-advanced information Ft+δ, where Fs is the σ-algebra generated by Brownian motion up to time s, s ≥ −δ, δ > 0 being a fixed constant. Our approach involves the forward integral and the Malliavin calculus fo...