论文标题
$ l_p $对频谱负lévy过程的最后一个零的最佳预测
$L_p$ optimal prediction of the last zero of a spectrally negative Lévy process
论文作者
论文摘要
鉴于频谱负Lévy流程$ x $ drifting to Infinity（启发在Shiryaev的早期想法（2002））中，我们有兴趣找到一个停止时间，以最大程度地减少$ l^p $ dance（$ p> 1 $），$ g $，最后一次$ x $是负的。与$ p = 1 $的情况相比，该解决方案要困难得多，在Baurdoux和Pedraza（2020b）中显示，该解决方案是最佳停止在$ x $超过恒定障碍的最佳状态。在此处处理的$ p> 1 $的情况下，我们证明解决此最佳预测问题等同于解决最佳的停止问题，从二维强的马尔可夫进程来解决，该过程将当前旅行的长度纳入了$ 0 $。我们表明，现在首次给出了最佳的停止时间，即$ x $超过非侵入和非负曲线，具体取决于当前旅行的长度，从$ 0 $ $ 0。我们进一步将最佳边界和值函数描述为函数子类内积分方程的非线性系统的唯一解。例如，考虑到带有指数跳跃的泊松工艺扰动的布朗运动和带有漂移的布朗运动的情况。
Given a spectrally negative Lévy process $X$ drifting to infinity, (inspired on the early ideas of Shiryaev (2002)) we are interested in finding a stopping time that minimises the $L^p$ distance ($p>1$) with $g$, the last time $X$ is negative. The solution is substantially more difficult compared to the case $p=1$, for which it was shown in Baurdoux and Pedraza (2020b) that it is optimal to stop as soon as $X$ exceeds a constant barrier. In the case of $p>1$ treated here, we prove that solving this optimal prediction problem is equivalent to solving an optimal stopping problem in terms of a two-dimensional strong Markov process that incorporates the length of the current excursion away from $0$. We show that an optimal stopping time is now given by the first time that $X$ exceeds a non-increasing and non-negative curve depending on the length of the current excursion away from $0$. We further characterise the optimal boundary and the value function as the unique solution of a non-linear system of integral equations within a subclass of functions. As examples, the case of a Brownian motion with drift and a Brownian motion with drift perturbed by a Poisson process with exponential jumps are considered.
