Markov processes

Kolmogorov-Pearson diffusions and jump -diffusion models with application to insurance

We introduce and study the class of Kolmogorov-Pearson diffusions or diffusions with jumps which have stationary distributions of Pearson type. They includes Ornstein-Uhlenbeck, Cox-Ingersoll-Ross processes which are widely used in financial econometrics and several others which have a heavy-tailed distributions such as inverted Gamma, Student or Fisher-Snedecor. Among the others we developed a new method of investigation the ruin problem for insurance jump-diffusion models.

Collaborators: Prof. Florin Avram, Prof. Alexei Kulik, Dr. Landy Rabenhasaina, Dr. Nenad Suvak, Dr. D. Denisov

Markov processes and eigenvalues of GUE

We study exit times of multidimensional Markov processes and construct conditional versions of these processes. We prove limit theorems which show that the resulting procesess are connected with eigenvalues of the Gaussian Unitary Ensemble (GUE).

Collaborators: Professor Sergei Foss, Professor Takis Konstantopoulos, Dr. Vitali Wachtel, Dr. D. Denisov

Cardiff Investigators

• Prof. N. Leonenko

Selected publications

• Avram, F., Leonenko, N.N., Suvak, N., (2011), Parameter estimation for Fisher-Snedecor diffusion, Statistics, 45, N1, 27-42
• Kulik, A.M., Leonenko, N.N. (2011) Ergodicity and mixing bounds for the Fisher-Snendecor diffusion, Bernoulli, submitted
• Leonenko, N.N., Suvak, N., (2010), Statistical inference for reciprocal gamma diffusion processes, J. Statist. Plann. Inference, 140, 30-51
• Leonenko, N.N., Suvak, N., (2010), Statistical inference for Student diffusion process, Stoch. Anal. Appl. 28, N6, 972-1002
• Avram, F., Leonenko, N.N., Suvak, N., (2010) On spectral properties and statistical analysis of Fisher-Snedecor diffusion, arXiv:1007.4909v1
• Avram, F., Leonenko, N.N., Rabenhasaina, L. (2009) Series expansions for the first passage distribution of Wong-Pearson jump-diffusions. Stoch. Anal. Appl. 27 , no. 4, 770-796.

• Denisov D., Wachtel V. (2011) Random Walks in Cones. Submitted. ArXiv:1110.1254
• Denisov D., Wachtel V. (2011) Ordered random walks with heavy tails.To appear in Elec. J. Probab. ArXiv:1103.4529.
• Denisov D., Wachtel V. (2010) Conditional limit theorems for ordered random walks. Elec. J. Probab. 10, 292-322. arXiv:0907.2854.
• Denisov D., Foss S., Konstantopoulos T. (2011) Limit theorems for a random directed slab graph. To appear in Ann. Appl. Probab. arXiv:1005.4806.