Viable insider markets

ABSTRACT

In this paper we investigate a logarithmic utility maximization problem of the terminal wealth for an insider portfolio, where the inside information consists of knowledge of some future values of the Brownian motion $B\left(t\right)$ driving the financial market. More specifically, we assume that at time t the insider has access to market information at least ${\epsilon }_t>0$ time units ahead of time. We consider two cases: the first case is when $t+{\epsilon }_t>T$ and the second case is when $t+{\epsilon }_t<T$.

The goal of this paper is to study when the market is viable, in the sense that the corresponding utility maximization problem admits a finite value.

In Case 1 we give a necessary condition for the viability of the market. We prove that the market is viable only if ${\int }_0^T\frac{1}{{\epsilon }_t}\mathrm{d}t<\mathrm{\infty }.$ In Case 2 we give a sufficient and necessary conditions for the viablity and we prove that the market is not viable for any choice of ${\epsilon }_t>0$.

KEYWORDS:

• Stochastic differential equation
• viable financial market
• logarithmic utility
• inside information
• Donsker delta functional
• Hida–Malliavin calculus
• forward integrals
• optimal insider control

MSC(2010):

• 60H10
• 60H40
• 60J65
• 91B55
• 91B70
• 91G80
• 93E20

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was carried out with support of the Norwegian Research Council, within the research project Challenges in Stochastic Control, Information and Applications (STOCONINF), project number 250768/F20.