We study optimal insider control problems, i.e., optimal control problems of stochastic systems where the controller at any time t, in addition to knowledge about the history of the system up to this time, also has additional information related to a future value of the system. Since this puts the associated controlled systems outside the context of semimartingales, we apply anticipative white noise analysis, including forward integration and Hida-Malliavin calculus to study the problem. Combining this with Donsker delta functionals, we transform the insider control problem into a classical (but parametrised) adapted control system, albeit with a non-classical performance functional. We establish a sufficient and a necessary maximum principle for such systems. Then we apply the results to obtain explicit solutions for some optimal insider portfolio problems in financial markets described by Itô-Lévy processes. Finally, in the Appendix, we give a brief survey of the concepts and results we need from the theory of white noise, forward integrals and Hida-Malliavin calculus.Keywords Optimal inside information control · Hida-Malliavin calculus · Donsker delta functional · Anticipative stochastic calculus · BSDE · Optimal insider portfolio Mathematics Subject Classification 60H40 · 60H07 · 60H05 · 60J75 · 60Gxx · 91G80 · 93E20 · 93E10 B Bernt Øksendal oksendal@math.uio.no Olfa Draouil

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

MSC(2010): 60H40, 60H07, 60H05, 60J75, 60J75, 60Gxx, 91G80, 93E20, 93E10Keywords: Optimal control, inside information, white noise, Hida-Malliavin calculus, Donsker delta functional, anticipative stochastic calculus, maximum principle, BSDE, optimal insider consumption and optimal insider portfolio under model uncertainty. AbstractWe study stochastic differential games of jump diffusions, where the players have access to inside information. Our approach is based on anticipative stochastic calculus, white noise, Hida-Malliavin calculus, forward integrals and the Donsker delta functional. We obtain a characterization of Nash equilibria of such games in terms of the corresponding Hamiltonians. This is used to study applications to insider games in finance, specifically optimal insider consumption and optimal insider portfolio under model uncertainty.We now explain this in more detail: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), jointly defined on a filtered probability space (Ω, F = {F t } t≥0 , P) satisfying the usual conditions. We assume that the inside information is of initial enlargement type. Specifically, we assume that the two inside filtrations H 1 , H 2 representing the information flows available to player 1 and player 2, respectively, have the formfor all t, where Y i is a given F T 0 -measurable random variable, for some fixed T 0 > T > t.Here the insider control process u(t) = (u 1 (t), u 2 (t)), where u i (t) is the control of player i; i=1,2. Thus we assume that the value at time t of our insider control process u i (t) is allowed to depend on both Y i and F t ; i = 1, 2. In other words, u i is assumed to be H i -adapted for i = 1, 2. Therefore they have the formFor simplicity (albeit with some abuse of notation) we will in the following write u i in stead of u i ; i = 1, 2. Consider a controlled stochastic processwhere u i (t) = u i (t, y i ) y i =Y i is the control process of insider i; i = 1, 2, and the (anticipating) stochastic integrals are interpreted as forward integrals, as introduced in [RV] (Brownian motion case) and in [DMØP1] (Poisson random measure case). A motivation for using forward integrals in the modelling of insider control is given in [BØ]. Let A i denote a given set of admissible H i −adapted controls u i of player i, with values inThe performance functional J i (u); u = (u 1 , u 2 ) of player i is defined by, writing y = (y 1 , y 2 ) and dy = dy 1 dy 2 ,x(t, y 1 , y 2 ), u 1 (t, y 1 ), u 2 (t, y 2 ), y)E[δ Y 1 ,Y 2 (y 1 , y 2 )|F t ]dt + g i (x(T, y 1 , y 2 ), y 1 , y 2 )E[δ Y 1 ,Y 2 (y 1 , y 2 )|F T ]}dy]; i = 1, 2.(1.4)A Nash equilibrium for the game (3.3)-(3.8) is a pairû = (û 1 ,û 2 ) ∈ A 1 × A 2 such that sup u 1 ∈A 1 J 1 (u 1 ,û 2 ) ≤ J 1 (û 1 ,û 2 ) (1.5) {eq2.10}

MSC(2010): 60H05; 60H07; 60H40; 60G57; 91B70; 93E20.Keywords: Stochastic delay equation, optimal insider control, Hida-Malliavin derivative, Donsker delta functional, white noise theory, stochastic maximum principles, time-advanced BSDE, optimal insider portfolio in a financial market with delay. AbstractWe use a white noise approach to study the problem of optimal inside control of a stochastic delay equation driven by a Brownian motion B and a Poisson random measure N . In particular, we use Hida-Malliavin calculus and the Donsker delta functional to study the problem.

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