  We study the small time path behavior of double stochastic integrals of the
form $\int_0^t(\int_0^rb(u) dW(u))^T dW(r)$, where $W$ is a $d$-dimensional
Brownian motion and $b$ is an integrable progressively measurable stochastic
process taking values in the set of $d\times d$-matrices. We prove a law of the
iterated logarithm that holds for all bounded progressively measurable $b$ and
give additional results under continuity assumptions on $b$. As an application,
we discuss a stochastic control problem that arises in the study of the
super-replication of a contingent claim under gamma constraints.
