We consider a new class of processes, called LG processes, defined as linear combinations of independent gamma processes. Their distributional and path-wise properties are explored by following their relation to polynomial and Dirichlet (B-) splines. In particular, it is shown that the density of an LG process can be expressed in terms of Dirichlet (B-) splines, introduced independently by Ignatov and Kaishev (1987, 1988, 1989a,b) and Karlin et al. (1986). We further show that the well known variance-gamma (VG) process, introduced by Madan and Seneta (1990), and the Bilateral Gamma (BG) process, recently considered by Kuchler and Tappe (2008) are special cases of an LG process. Following this LG interpretation, we derive new (alternative) expressions for the VG and BG densities and consider their numerical properties. The LG process has two sets of parameters, the B-spline knots and their multiplicities, and offers further flexibility in controlling the shape of the Levy density, compared to the VG and the BG processes. Such flexibility is often desirable in practice, which makes LG processes interesting for financial and insurance applications.ududMultivariate LG processes are also introduced and their relation to multivariate Dirichlet and simplex splines is established. Expressions for their joint density, the underlying LG-copula, the characteristic, moment and cumulant generating functions are given. A method for simulating LG sample paths is also proposed, based on the Dirichlet bridge sampling of Gamma processes, due to Kaishev and Dimitrova (2009). A method of moments for estimation of the LG parameters is also developed. Multivariate LG processes are shown to provide a competitive alternative in modelling dependence, compared to the various multivariate generalizations of the VG process, proposed in the literature. Application of multivariate LG processes in modelling the joint dynamics of multiple exchange rates is also considered.
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机译：我们考虑一类新的过程，称为LG过程，定义为独立伽玛过程的线性组合。通过遵循它们与多项式和Dirichlet（B-）样条的关系来探索它们的分布和路径特性。特别是，显示出LG过程的密度可以用Dirichlet（B-）样条表示，由Ignatov和Kaishev（1987，1988，1989a，b）和Karlin等人独立引入。 （1986）。我们进一步表明，由Madan和Seneta（1990）引入的众所周知的方差伽玛（VG）过程以及最近由Kuchler和Tappe（2008）考虑的双边Gamma（BG）过程都是LG过程的特例。根据LG的这种解释，我们得出VG和BG密度的新（替代）表达式，并考虑它们的数值特性。 LG工艺具有两组参数，即B样条结及其多重性，与VG和BG工艺相比，在控制Levy密度的形状方面提供了更大的灵活性。在实践中通常需要这种灵活性，这使得LG过程对于金融和保险应用很有趣。 ud ud也介绍了多元LG过程，并建立了它们与多元Dirichlet和单纯形样条的关系。给出了它们的联合密度，潜在的LG系，特征，矩和累积量生成函数的表达式。由于Kaishev和Dimitrova（2009），还提出了一种基于伽玛过程的Dirichlet桥采样的LG采样路径模拟方法。还开发了一种估计LG参数的矩方法。与文献中提出的VG流程的各种多元概括相比，多元LG流程显示出在建模依赖性方面提供了竞争性选择。还考虑了多元LG流程在建模多种汇率的联合动态中的应用。
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