二重随机积分的小时间路径性质及其在随机控制中的应用
Small time path behavior of double stochastic integrals and applications to stochastic control
摘要：
研究了$\int_0^t(\int_0^rb(u) dW(u))^T dW(r)$型二重随机积分的小时间路径行为，其中$W$是$d$维布朗运动，$b$是取值于$d\times d$ -矩阵集合的可积渐进可测随机过程.我们证明了一个法律的重对数，适用于所有有界渐进可测$b$，并给出额外的结果连续性假设下，$b$。作为应用，我们讨论了伽玛约束下未定权益超复制研究中的一个随机控制问题。
英文摘要：
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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