论文标题
具有完全单调跳跃的Lévy过程的上流：光谱理论方法
Suprema of Lévy processes with completely monotone jumps: spectral-theoretic approach
论文作者
论文摘要
我们研究了在研究具有单方面屏障的完全单调跳跃的一维lévy工艺中产生的非自动选择运算符的光谱理论特性。在没有进一步的假设的情况下，我们为过渡密度的双变量拉普拉斯变换提供了不可或缺的表达式，$ p_t^+（x，y）$ in $（0，\ infty）$，在较小的规律性条件下，为相应的过渡运营商$ p_t^+$提供了广义的特征函数扩展。假设特征指数的适当增长，我们证明了过渡密度$ p_t^+（x，y）$的广义特征功能扩展。在类似的条件下，我们还显示了intimum和the the the Euntimum and Ecremum函数的累积分布函数的积分公式$ \ usewissline {x} _t $和$ \ overline {x} _t $。我们的结果所涵盖的过程类别包括许多稳定且稳定的Lévy过程，以及许多带有布朗组件的过程。我们的结果恢复了经典风险过程的已知表达式，并为Lévy过程的其他一些简单示例提供了类似的积分公式。
We study spectral-theoretic properties of non-self-adjoint operators arising in the study of one-dimensional Lévy processes with completely monotone jumps with a one-sided barrier. With no further assumptions, we provide an integral expression for the bivariate Laplace transform of the transition density $p_t^+(x, y)$ of the killed process in $(0, \infty)$, and under a minor regularity condition, a generalised eigenfunction expansion is given for the corresponding transition operator $P_t^+$. Assuming additionally appropriate growth of the characteristic exponent, we prove a generalised eigenfunction expansion of the transition density $p_t^+(x, y)$. Under similar conditions, we additionally show integral formulae for the cumulative distribution functions of the infimum and supremum functionals $\underline{X}_t$ and $\overline{X}_t$. The class of processes covered by our results include many stable and stable-like Lévy processes, as well as many processes with Brownian components. Our results recover known expressions for the classical risk process, and provide similar integral formulae for some other simple examples of Lévy processes.
