On the convex hull of symmetric stable processes

Abstract

Let α ∈ ( 1 , 2 ] \alpha \in (1,2] and X X be an R d \mathbb R^d -valued symmetric α \alpha -stable Lévy process starting at 0 0 . We consider the closure S t S_t of the path described by X X on the interval [ 0 , t ] [0,t] and its convex hull Z t Z_t . The first result of this paper provides a formula for certain mean mixed volumes of Z t Z_t and in particular for the expected first intrinsic volume of Z t Z_t . The second result deals with the asymptotics of the expected volume of the stable sausage Z t + B Z_t+B (where B B is an arbitrary convex body with interior points) as t → 0 t\to 0 . For this we assume that X X has independent components.