Non-parametric kernel estimation for symmetric Hawkes processes. Application to high frequency financial data

We define a numerical method that provides a non-parametric estimation of the kernel shape in symmetric multivariate Hawkes processes. This method relies on second order statistical properties of Hawkes processes that relate the covariance matrix of the process to the kernel matrix. The square root of the correlation function is computed using a minimal phase recovering method. We illustrate our method on some examples and provide an empirical study of the estimation errors. Within this framework, we analyze high frequency financial price data modeled as 1D or 2D Hawkes processes. We find slowly decaying (power-law) kernel shapes suggesting a long memory nature of self-excitation phenomena at the microstructure level of price dynamics.Comment: 6 figure

Image-guided Cryotherapy for Musculoskeletal Tumors

Background: This article represents a review about the use of image-guided cryotherapy in the treatment of musculoskeletal tumor lesions. Cryotherapy is able to induce a lethal effect on cancer cells through direct and indirect mechanisms. In this manuscript, we combined our experience with that of other authors who have published on this topic in order to provide indications on when to use cryotherapy in musculoskeletal oncology. Discussion: Image-Guided percutaneous cryotherapy is a therapeutic method now widely accepted in the treatment of patients with musculoskeletal tumors. It can be used both for palliative treatments of metastatic bone lesions and for the curative treatment of benign bone tumors such as osteoid osteoma or osteoblastoma. In the treatment of bone metastases, cryotherapy plays a major role in alleviating or resolving disease-related pain but it has also been demonstrated that it can have a role in local disease control. In recent years, the use of cryotherapy have also expanded for the treatment of both benign and malignant soft tissue tumors. Conclusion: Percutaneous cryotherapy can be considered a safe and effective technique in the treatment of benign and malignant musculoskeletal tumors. Cryotherapy can be considered the first option in benign tumor lesions such as osteoid osteoma and a valid alternative to radiofrequency ablation. In the treatment of painful bone metastases, it must be considered secondarily to other standard treatments (radiotherapy, bisphosphonate therapy and chemotherapy) where they are no longer effective in controlling the disease or when they cannot be repeated (for example radiotherapy)