## A New Characterization of Random Times for Specifying Information Delay

J. Math. Sci. Univ. Tokyo
Vol. 20 (2013), No. 1, Page 147–170.

Adachi, Takanori ; Miura, Ryozo ; Nakagawa, Hidetoshi
A New Characterization of Random Times for Specifying Information Delay
We introduce a stochastic process called a follower process consisting of a non-decreasing sequence of random times $f_t$ whose values do not exceed $t$. It was originally introduced for representing information delay in structural credit risk models. The follower process is an extension of a time change process introduced by Guo, Jarrow and Zeng in the sense that each component of the follower process is not required to be a stopping time. We introduce a class of follower processes called idempotent, which contains natural examples including follower processes driven by renewal processes. We show that any idempotent follower process is hard to be an example of time change processes. We define a filtration modulated by the follower process and show that it is a natural extension of the continuously delayed filtration that is the filtration modulated by the time change process. We show that conditional expectations given idempotent follower filtrations have some Markov property in a binomial setting, which is useful for pricing defaultable financial instruments.