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

Abstract

We study the small time path behavior of double stochastic integrals of othe form f(0)(t)(f(0)(r) b(u) dW(u))(T) dW(r), where W is ad-dimensional Brownian motion and b is an integrable progressively measurable stochastic process taking values in the set of d x 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.