Appl Math Optim 52:167–181 (2005)
2005 Springer Science+Business Media, Inc.
A General Stochastic Calculus Approach to Insider Trading
and Bernt Øksendal
Department of Mathematics, University of Bologna,
Piazza di Porta S. Donato 5, I-40127 Bologna, Italy
Center of Mathematics for Applications (CMA),
Department of Mathematics, University of Oslo,
Box 1053 Blindern, N-0316 Oslo, Norway
Norwegian School of Economics and Business Administration,
Helleveien 30, N-5045 Bergen, Norway
Abstract. The purpose of this paper is to present a general stochastic calculus
approach to insider trading. We consider a market driven by a standard Brownian
motion B(t) on a ﬁltered probability space (, F,
, P) where the coefﬁcients
are adapted to a ﬁltration G =
, with F
for all t ∈ [0, T ], T > 0
being a ﬁxed terminal time. By an insider in this market we mean a person who
has access to a ﬁltration (information) H =
which is strictly bigger than
the ﬁltration G =
. In this context an insider strategy is represented by an
-adapted process ϕ(t) and we interpret all anticipating integrals as the forward
integral deﬁned in  and .
We consider an optimal portfolio problem with general utility for an insider
with access to a general information H
and show that if an optimal insider
(t) of this problem exists, then B(t) is an H
-semimartingale, i.e. the
enlargement of ﬁltration property holds. This is a converse of previously known
results in this ﬁeld. Moreover, if π
exists we obtain an explicit expression in terms
for the semimartingale decomposition of B(t) with respect to H
. This is a
generalization of results in ,  and .
AMS Classiﬁcation. Primary 60HXX, Secondary 91B28.
Key Words. Forward integral, Skorohod integral, Wick product, Insider trading,
This research was supported by the University of Bologna. Funds for selected research topics.