Small time path behavior of double stochastic integrals and applications to stochastic control

We study the small time path behavior of double stochastic integrals of the form ${\int }_0^t({\int }_0^{r}b(u)dW(u){)}^{T}dW(r)$, 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.