Initial Enlargement of Filtrations and Entropy of Poisson Compensators

Abstract

Let μ be a Poisson random measure, let F be the smallest filtration satisfying the usual conditions and containing the one generated by μ, and let G be the initial enlargement of F with the σ -field generated by a random variable G. In this paper, we first show that the mutual information between the enlarging random variable G and the σ -algebra generated by the Poisson random measure μ is equal to the expected relative entropy of the G-compensator relative to the F-compensator of the random measure μ. We then use this link to gain some insight into the changes of Doob-Meyer decompositions of stochastic processes when the filtration is enlarged from F to G. In particular, we show that if the mutual information between G and the σ -algebra generated by the Poisson random measure μ is finite, then every squareintegrable F-martingale is a G-semimartingale that belongs to the normed space S 1 relative to G.

Arbitrage and utility maximization in market models with an insider

“…The filtration F is generated by the two independent standard Poisson processes N 1 and N 2 . Any martingale density has the form (7) where now α 1 and α 2 are positive predictable integrable processes satisfying θ α 1 t + (1 − θ )α 2 t = 1, P − a.s. Therefore,…”

“…Insider trading under initial enlargement of filtration has been the object of interest of many papers, including but by no means limited to [12,3,16,6,7,4,32,17]. The majority of these papers work in a complete market setting and are concerned with the question of additional utility of the insider; they find that when the variable G is F T -measurable and not purely atomic, this additional utility is often infinite.…”

Anticipative information in a Brownian-Poissonmarket: the binary information

“…In the seminal paper [3], it is shown that if the dynamics of the risky asset do not include the discontinuous part N , then the additional gain is given by the entropy of the random variable G. After that, much progress has been made in the analysis of the additional information in the Brownian case, see [4,5,6] for the main references. The research on the Poisson process in the initial enlargement framework started with [7,8] in which the existence of a compensator is analyzed. Although they consider the entropy, the additional gain of an informed G-agent in the optimal portfolio problem has not been studied.…”