Optimal insider control and semimartingale decompositions under enlargement of filtration

Abstract

We combine stochastic control methods, white noise analysis and Hida-Malliavin calculus applied to the Donsker delta functional to obtain explicit representations of semimartingale decompositions under enlargement of filtrations. Some of the expressions are more explicit than previously known. The results are illustrated by examples.Keywords: Enlargement of filtration, Semimartingale decomposition, Optimal inside information control, Hida-Malliavin calculus, Donsker delta functional.MSC (2010): 60H40, 60H07, 60H05, 60J75, 60G48, 91G80, 93E20 IntroductionThe purpose of this paper is twofold:• We introduce a new approach to enlargement of filtration problems, based on combining several optimal control methods.• We show that this approach can in some cases give more explicit results than known before.The system we consider, is described by a stochastic differential equation driven by a Brownian motion B(t) and an independent compensated Poisson random measureÑ(dt, dζ) = N(dt, dζ) − ν(dζ)dt, ν being the Lévy measure of the Poisson random measure N. The processes are jointly defined on a filtered probability space (Ω, F = {F t } t≥0 , P) satisfying the usual conditions where Ω = S ′ (R) and P is the Gaussian measure on S ′ (R). Here, and throughout the paper, F t denotes the sigma-algebra generated by {B(s)} s≤t and {N(s, ·)} s≤t . Throughout this paper we assume that the inside information is of initial enlargement type. Specifically, we assume that the inside filtration H has the form

Kyle–back’s Model With a Random Horizon

“…In this direction we find, e.g. Karatzas & Pikovsky (1996), Amendiger et al (1998), Grorud & Pontier (1998, Imkeller et al (2001), Corcuera et al (2004), Biagini & Øksendal (2005Biagini & Øksendal ( , 2006, Kohatsu-Higa (2007), Di Nunno et al (2006, Draouil & Øksendal (2016), Enrst et al (2017). In some cases, the impact of the insider strategy affects the stock price dynamics in the sense that these dynamics are dependent on the insider strategy itself; see, e.g., Di Nunno et al (2008).…”

A White Noise Approach to Insider Trading

“…This result is a special case of semimartingale decomposition results for Lévy processes in [DrØ2]. For information about enlargement of filtration in general, see [JY] and [J] and the references therein.…”

“…Combining this result with the results of [PK] and [BØ] given in the Introduction, we get the following result of independent interest. It is a special case of results in [DrØ2]:…”

A Triple Comparison between Anticipating Stochastic Integrals in Financial Modeling

“…While any of these anticipating stochastic integrals guarantees the well-posedness of the problem under study, the financial consequences of each choice might be very different. At this moment it is important to remark that this sort of problem has been approached before through the use of the Russo-Vallois forward integral [4,7,8,9,10,19,22]. On one hand, the justification of such a choice has frequently been made based on technical arguments.…”