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

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

2006-02-21

arxiv.org/abs/math/0602453v1

Annals of Applied Probability 2005, Vol. 15, No. 4, 2472-2495

Mathematics

Probability

Published at http://dx.doi.org/10.1214/105051605000000557 in the Annals of Applied Probability (http://www.imstat.org/aap/) by

Scientific paper

10.1214/105051605000000557

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.

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