A Donsker Delta Functional Approach to Optimal Insider Control and Applications to Finance

Abstract

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

Optimal insider control and semimartingale decompositions under enlargement of filtration

“…The advantages with working with the Donsker delta functional, rather than the regular conditional distribution, include the following (see e.g. [DØ1] for details and examples):…”

“…We refer to [DØ1] and the references therein for more details in this example. Consider the case when Y is a first order chaos random variable of the form…”

“…where D t and D t,ζ denote the Hida-Malliavin derivatives with respect to B(·) andÑ (·, ·), respectively. We refer to [DØ1] for more information about (S) * , white noise theory and Hida-Malliavin derivatives. As pointed out in [DØ1], conditions (1.2) hold automatically if Y has the form Y = for some deterministic functions ϕ, ψ such that ϕ ∈ L 2 [m] and ψ ∈ L 2 [m × ν], where m is Lebesgue measure on [0, T 0 ], and we assume that the Lévy measure ν satisfies the condition R ζ 2 dν(ζ) < ∞.…”

Optimal insider control of stochastic partial differential equations

“…To study this problem we adapt the technique of the paper [DØ1] to the stochastic Volterra equation (SVE) and we combine this with the method for optimal control of SVE developed in [AØ]. We first recall briefly the definition and basic properties of the Donsker delta functional: For example, consider the special case when Z is a first order chaos random variable of the form…”

Viable insider markets

“…The problem of insider trading is studied in many works for example [A,AIS,AI,BØ,C,DMØP2,DØ1,H,IPW,PK]. This type of problems is related to the enlargement of filtration [Ja, J, Je, M].…”