Talks

  • Lucia Caramellino (Università di Roma "Tor Vergata", Italy)

Title: Convergence of a Hybrid Numerical Scheme for Pricing Options

Abstract: We first study the rate of weak convergence of Markov chains to diffusion processes under quite general assumptions and we apply the convergence analysis to a multiple jumps tree approximation of the CIR process. We then combine the Markov chain approach with other numerical techniques in order to handle the different components in jump-diffusion coupled models. We study the analytical speed of convergence of this hybrid approach and apply the results to a tree-finite difference approximation of the Heston and Bates models.


  • Nikolaos Limnios (University of Technology of Compiegne, Sorbonne University, France)

Title: Functional Asymptotic Results for Near-Critical Branching Processes

Abstract: We consider discrete-time and continuous-time Markov branching processes and prove diffusion approximation results in the near critical case, where the later one is considered in semi-Markov random environment. In that case, we obtain average and diffusion approximation results.

In one hand, we provide new proofs to the known results Feller-Jirina and Jagers theorems, in the fixed environment case, and we provide conditions for diffusion approximation. In the other hand, we propose a continuous-time Markov branching process with semi-Markov random environments and obtain diffusion approximation results. We present also an averaging result. The main steps of the proofs, obtained by singular perturbation of stochastic operators, are provided.



  • Barbara Rüdiger (Bergische Universität Wuppertal, Germany)

Title: The Boltzmann –Enskog process

Abstract: The theory of SDEs with Poisson noise is used here to identify the Boltzmann –Enskog –Process. Its dynamic describes the space and velocity evolution of a particle of a rarified gas which density evolves according to the Boltzmann – Enskog equation. This random dynamic is identified by a stochastic process solving a SDE, for which the corresponding Kolmogorov equation is given by the Boltzmann –Enskog equation. It turns out that this is the solution of a Mc Kean –Vlasov type SDE with Poisson noise: its compensator is determined by the density solving the Boltzmann –Enskog equation.

The talk is based on joint results with S. Albeverio and P. Sundar, as well as , M. Friesen and P. Sundar


  • Francesco Russo (ENSTA Paris | Institut Polytechnique de Paris, France)

Title: Weak Dirichlet processes with jumps and solutions of path-dependent SDEs with distributional drift

Abstract: In this talk we revisit the stochastic calculus for weak Dirichlet processes which are the natural extension of semimartingale with jumps. A weak Dirichlet process X is the sum of a local martingale M and a martingale ortogonal process A in the sense that [A,N] = 0 for every continuous local martingale N. We remark that if [A] = 0 then X is a Dirichlet process. The notion of Dirichlet process is not very suitable in the jump case since in this case A is forced to be continuous. The notion of weak Dirichlet process is also naturally related to the one of stochasticalled controlled process (in the sense of rough paths).

In the second part of the talk we focus on the example of a solution to a path-dependent SDE with jumps and possible distributional drift. The talk is based on a joint papers with E. Bandini (Bologna).


  • Radomyra Shevchenko (Max Planck Institute, Germany)

Title: Estimation questions for fractional Ornstein-Uhlenbeck type SDEs

Abstract: This talk gives an overview over several results concerning drift estimators of Ornstein-Uhlenbeck type equations with a periodic drift driven by a fractional long range dependent process such as fractional Brownian motion or the Rosenblatt process. The main challenge in dealing with such equations is that neither the driving processes nor the solutions are semimartingales, and therefore defining estimators and studying their asymptotics requires tools outside the classical Itô calculus. While in the last decades several useful theories such as Malliavin calculus have been developed, most problems in mathematical statistics for fractional processes remain case-to-case studies. In the talk we will consider parametric and nonparametric estimators for continuous time observations adapted to different sets of assumptions and defined via either Skorokhod or Riemann-Stieltjes integrals. (Based on joint work with C.A. Tudor and J.H.C. Woerner).