论文标题
正面希尔伯特·史克米特操作员的纯粹纯种过程
Affine pure-jump processes on positive Hilbert-Schmidt operators
论文作者
论文摘要
我们展示了积极自我伴侣希尔伯特·史克米特（Hilbert-Schmidt Operators）锥体中的一系列仿射马尔可夫过程。此类过程非常适合无限尺寸随机波动率模型。我们认为的过程类别是在Cuchiero等人研究的阳性半明确和对称矩阵的空间中仿射过程的无限尺寸类似物。 [安。应用。概率。 21（2011）397-463]。与有限维度一样，我们构建的过程允许根据状态线性线性仿射，并受到跳跃措施的跳跃，该跳跃度量依赖于状态。但是，由于Hilbert-Schmidt阳性阳性的无限尺寸锥体具有空内部装置，因此我们不考虑扩散项。空旷的内部还需要一种新的证明存在方法：我们不使用标准定位技术，而是采用了在Dörsek和Teichmann引入的广义的Feller Semigroup [Arxiv 2010]中的理论，并在Cuchiero和Teichmann进一步开发[Evolution of Evolution of Evolution of Evolution of Evolution equontion equation of Evolution equication equolution equication of Evolution equication equontion equication extions（2020）]。我们的方法需要对所涉及的跳跃措施的第二刻条件，因此，我们获得了仿射过程的第一和第二矩的明确公式。
We show the existence of a broad class of affine Markov processes in the cone of positive self-adjoint Hilbert-Schmidt operators. Such processes are well-suited as infinite dimensional stochastic volatility models. The class of processes we consider is an infinite dimensional analogue of the affine processes in the space of positive semi-definite and symmetric matrices studied in Cuchiero et al. [Ann. Appl. Probab. 21 (2011) 397-463]. As in the finite dimensional case, the processes we construct allow for a drift depending affine linearly on the state, as well jumps governed by a jump measure that depends affine linearly on the state. However, because the infinite-dimensional cone of positive self-adjoint Hilbert-Schmidt operators has empty interior, we do not consider a diffusion term. This empty interior also demands a new approach to proving existence: instead of using standard localisation techniques, we employ the theory on generalized Feller semigroups introduced in Dörsek and Teichmann [arXiv 2010] and further developed in Cuchiero and Teichmann [Journal of Evolution Equations (2020)]. Our approach requires a second moment condition on the jump measures involved, consequently, we obtain explicit formulas for the first and second moments of the affine process.
