Stochastic Calculus for a Time-Changed Semimartingale and the Associated Stochastic Differential Equations
Stochastic Calculus for a Time-Changed Semimartingale and the Associated Stochastic Differential Equations
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DOI：
10.1007/s10959-010-0320-9
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发表时间：
2011-09-01
影响因子：
0.8
通讯作者：
Kobayashi, Kei
中科院分区：
文献类型：
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作者：
Kobayashi, Kei
结果表明，在半鞅和时变的一定条件下，时变半鞅驱动的任意随机积分都是原半鞅驱动的时变随机积分。作为直接结果，导出了 It 公式的特殊形式。当标准布朗运动是原始半鞅时，由带有漂移的布朗运动驱动的经典随机微分方程扩展到更大类涉及连续路径的时间变化的随机微分方程。建立了这个新类中线性方程通解的形式，然后考虑了一些类似于经典方程的例子。通过这些例子，新类中随机微分方程的每个系数都被赋予了意义。新功能是常见的漂移项与与时间变化相关的项共存。
It is shown that under a certain condition on a semimartingale and a time-change, any stochastic integral driven by the time-changed semimartingale is a time-changed stochastic integral driven by the original semimartingale. As a direct consequence, a specialized form of the It formula is derived. When a standard Brownian motion is the original semimartingale, classical It stochastic differential equations driven by the Brownian motion with drift extend to a larger class of stochastic differential equations involving a time-change with continuous paths. A form of the general solution of linear equations in this new class is established, followed by consideration of some examples analogous to the classical equations. Through these examples, each coefficient of the stochastic differential equations in the new class is given meaning. The new feature is the coexistence of a usual drift term along with a term related to the time-change.