# Dark Markets: Asset Pricing and Information Transmission in over-the-Counter Markets

By Darrell Duffie | Go to book overview

APENDIX B
Counting Processes

This appendix reviews intensity-based models of counting processes. Bremaud (1981) is a standard source.

All properties below are with respect to a probability space (Ω, F, ℙ) and a given filtration {Ft: t0} satisfying the usual conditions unless otherwise indicated. We say that some X : Ω. × [0, ∞) is adapted if, for each time t, the function X(·, t): Ω. ℝ, also denoted Xt or X(t), is Ft -measurable. For market applications, Ft corresponds to the information held by a given set of agents at time t. To say that a process X is adapted can be interpreted as stating that Xt is observable at time t, or could be chosen by agents at time t, on the basis of the information represented by Ft.

A process Y is predictable if Y : Ω × [0,∞) is measurable with respect to the σ-algebra on Ω × [0, ∞) generated by the set of all leftcontinuous adapted processes. The idea is that one can [foretell] Yt based on all of the information available up to, but not including, time t. Of course, any left-continuous adapted process is predictable, as is, in particular, any continuous process.

A counting process N is defined via an increasing sequence {T0, T1,…} of random variables valued in [0,∞], with T0 = 0 and with Tn<Tn+1 whenever Tn < ∞, according to

where we define Nt = +∞ if tlimnTn. We may treat Tn as the nth jump time of N, and Nt as the number of jumps that have occurred up to and including time t. The counting process is nonexplosive if lim Tn = +∞ almost surely.

Definitions of [intensity] vary slightly from place to place. One may refer to section II.3 of Brémaud (1981), in particular theorems T8 and T9, to compare other definitions of intensity with the following. Let λ be a

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