This document is the author's final manuscript accepted version of the journal article, incorporating any revisions agreed during the peer review process. Some differences between this version and the published version may remain. You are advised to consult the publisher's version if you wish to cite from it. Dynamic Markov bridges motivated by models of insider tradingAbstract Given a Markovian Brownian martingale , we build a process which is a martingale in its own filtration and satisfies 1 = 1 . We call a dynamic bridge, because its terminal value 1 is not known in advance. We compute explicitly its semimartingale decomposition under both its own filtration ℱ and the filtration ℱ , jointly generated by and . Our construction is heavily based on parabolic PDE's and filtering techniques. As an application, we explicitly solve an equilibrium model with insider trading, that can be viewed as a non-Gaussian generalization of Back and Pedersen's [3], where insider's additional information evolves over time.

An optimal investment problem is solved for an insider who has access to noisy information related to a future stock price, but who does not know the stock price drift. The drift is filtered from a combination of price observations and the privileged information, fusing a partial information scenario with enlargement of filtration techniques. We apply a variant of the Kalman-Bucy filter to infer a signal, given a combination of an observation process and some additional information. This converts the combined partial and inside information model to a full information model, and the associated investment problem for HARA utility is explicitly solved via duality methods. We consider the cases in which the agent has information on the terminal value of the Brownian motion driving the stock, and on the terminal stock price itself. Comparisons are drawn with the classical partial information case without insider knowledge. The parameter uncertainty results in stock price inside information being more valuable than Brownian information, and perfect knowledge of the future stock price leads to infinite additional utility. This is in contrast to the conventional case in which the stock drift is assumed known, in which perfect information of any kind leads to unbounded additional utility, since stock price information is then indistinguishable from Brownian information.

This paper studies the equilibrium pricing of asset shares in the presence of dynamic private information. The market consists of a risk-neutral informed agent who observes the firm value, noise traders, and competitive market makers who set share prices using the total order flow as a noisy signal of the insider's information. I provide a characterization of all optimal strategies, and prove existence of both Markovian and non Markovian equilibria by deriving closed form solutions for the optimal order process of the informed trader and the optimal pricing rule of the market maker. The consideration of non Markovian equilibrium is relevant since the market maker might decide to re-weight past information after receiving a new signal. Also, I show that a) there is a unique Markovian equilibrium price process which allows the insider to trade undetected, and that b) the presence of an insider increases the market informational efficiency, in particular for times close to dividend payment. * I benefited from helpful comments from Peter Bank, Rene Carmona, Christian Julliard, Dmitry Kramkov, Michael Monoyios, Andrew Ng, Bernt Øksendal and seminar and workshop participants at 4th Oxford -Princeton Workshop,

We consider an equilibrium modelà la Kyle-Back for a defaultable claim issued by a given firm. In such a market the insider observes continuously in time the value of firm, which is unobservable by the market maker. Using the construction of a dynamic Bessel bridge of dimension 3 in [5], we provide the equilibrium price and the optimal insider's strategy. As in [3], the information released by the insider while trading optimally makes the default time predictable in market's view at the equilibrium. We conclude the paper by comparing the insider's expected profits in the static and dynamic private information case. We also compute explicitly the value of insider's information in the special cases of a defaultable stock and a bond.

Given a deterministically time-changed Brownian motion Z starting from 1, whose timechange V (t) satisfies V (t) > t for all t > 0, we perform an explicit construction of a process X which is Brownian motion in its own filtration and that hits zero for the first time at V (τ ), where τ := inf{t > 0 : Z t = 0}. We also provide the semimartingale decomposition of X under the filtration jointly generated by X and Z. Our construction relies on a combination of enlargement of filtration and filtering techniques. The resulting process X may be viewed as the analogue of a 3-dimensional Bessel bridge starting from 1 at time 0 and ending at 0 at the random time V (τ ). We call this a dynamic Bessel bridge since V (τ ) is not known in advance. Our study is motivated by insider trading models with default risk, where the insider observes the firm's value continuously on time. The financial application, which uses results proved in the present paper, has been developed in the companion paper [6]. * LAGA, University Paris 13, and CREST, campi@math.univ-paris13.fr.