Stochastic calculus for convoluted L\'{e}vy processes
Stochastic calculus for convoluted L\'{e}vy processes
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DOI：
10.3150/07-bej115
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发表时间：
2008-05
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影响因子：
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通讯作者：
Christian Bender;T. Marquardt
中科院分区：
文献类型：
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作者：
Christian Bender;T. Marquardt
我们建立了一个随机演算，它是由一个纯跳跃、零期望的L过程与Volterra型核卷积而成的。这类过程包含，例如，马夸特研究的分数L过程[Bernoulli12(2006)1090-1126]。我们介绍的积分是Skorokhod积分。尽管如此，我们避免了Malliavin微积分和白噪声分析中的技术细节，给出了度量变化下基于期望的初步定义。作为主要结果，我们导出了一个将卷积和跳跃引起的不同贡献从记忆中分离出来的公式。
We develop a stochastic calculus for processes which are built by convoluting a pure jump, zero expectation L\'{e}vy process with a Volterra-type kernel. This class of processes contains, for example, fractional L\'{e}vy processes as studied by Marquardt [Bernoulli 12 (2006) 1090--1126.] The integral which we introduce is a Skorokhod integral. Nonetheless, we avoid the technicalities from Malliavin calculus and white noise analysis and give an elementary definition based on expectations under change of measure. As a main result, we derive an It\^{o} formula which separates the different contributions from the memory due to the convolution and from the jumps.