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 ∫ t 0 ( ∫ r 0 b(u)dW (u)) dW (r), where W is a d-dimensional Brownian motion and b an integrable progressively measurable stochastic process taking values in the set of d× dmatrices. 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.