Dynamic Markov bridges motivated by models of insider trading

Abstract

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.

Equilibrium model with default and dynamic insider information

“…Direct calculations give that, on [τ > 1], dR t = Zt−Rt V (t)−t dt + dB t , thus we can apply the results in Back and Pedersen [2] (see also [4], Proposition 3.2) to conclude that R t goes to Z 1 as t ↑ 1 a.s. on the set of non-default [τ > 1]. To deduce from it that X t → Z 1 a.s. on [τ > 1] when t ↑ 1, we have to show …”

“…Since this paper deals with a Kyle-Back equilibrium model with default and gradually revealing information, it could also be viewed as a generalization of Back and Pedersen [2] and Campi, Ç etin and Danilova [4] to a financial market with default.…”

Financial equilibrium with asymmetric information and random horizon

“…This is due to the fact that the announcement comes as a surprise even for the insider and therefore, she is not able to construct a bridge of random length τ for the demand process in order for the prices to converge to Γ (cf. the bridge construction in [7]). However, in equilibrium, she trades in such a way that the price process conditioned on no announcement, i.e., P , converges to Γ .…”

“…In line with the current literature, we assume that the price chosen by the market makers is a deterministic function of some process Y , which solves dY t = w(t, Y t ) dX t + b(t, Y t ) dt for some weighting function w and drift function b chosen by the market makers (see e.g. [1,4,7] among others for the use of a weighting function in the construction of the market makers' signal).…”

“…In this extension, the total demand of the noise traders is given by a Brownian motion and the future payoff of the asset has a general continuous distribution, while the informed trader's private information is still static. Kyle's model and its continuous-time extension by Back have been further extended to allow multiple informed traders [11,3], to include default risk [6] or to the case when the single informed trader receives a continuous signal as private information [4,7].…”