We study the small time path behavior of double stochastic integrals of the form f 0 t (f 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 × 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.
All Science Journal Classification (ASJC) codes
- Statistics and Probability
- Statistics, Probability and Uncertainty
- Double stochastic integrals
- Hedging under gamma constraints
- Law of the iterated logarithm
- Stochastic con-trol