Alexander Sokol

(Department of Mathematical Sciences, University of Copenhagen)

(Department of Mathematical Sciences, University of Copenhagen)

Thiele Seminar

Wednesday, 26 September, 2012, at 14:15, in D02 (1531-015)

Description:

In the first part of the talk, we consider the question of when an exponential martingale is a uniformly integrable martingale. A classical result of Novikov states that for a continuous local martingale $M$, the exponential martingale is a uniformly integrable martingale if $\exp(\frac{1}{2}[M]_\infty)$ is integrable, and the constant $\frac{1}{2}$ is optimal. We observe that earlier theorems by Lépingle & Mémin yield Novikov-type results as corollaries, and identify optimal constants in these criteria under certain conditions on the jumps. We also give elementary proofs of slight extensions of the Novikov-type criteria obtained earlier. Finally, we show how to apply exponential martingales to the construction of various types of counting processes with stochastic intensities, for example intensities depending on a diffusion.

Organised by: The T.N. Thiele Centre

Contact person: Jan Pedersen