10:30-11:30 José Luis Pérez (CIMAT)
Title: Periodic strategies in optimal execution with multiplicative price impactAbstract: We consider the so-called optimal execution problem in algorithmic trading, which is the problem faced by an investor who has a large number of stock shares to sell over a given time horizon and whose actions have an impact on the stock price. We provide a generalization of the model of Zervos and Guo by assuming that the investor can only sell at the arrival times of an independent Poisson process. In particular, we develop and study a price model that presents the stochastic dynamics of a geometric Brownian motion and incorporates a log-linear effect of the investor’s transactions. We then formulate the optimal execution problem as a stochastic control problem with periodic strategies. In particular, we derive an explicit solution to the problem if the time horizon is infinite.

11:45-12:45 Irmina Czarna (University of Wroclaw)
Title: Multi-refracted Lévy risk processes
Abstract: We consider multi-refracted Lévy risk process whose dynamics change by subtracting off a fixed linear drifts whenever the process is above certain levels. Formally, we define a multi-refracted Lévy risk process as a unique strong solution of the SDE (for $k \geq 1$): $$\mathrm{d}U_k(t) = \mathrm{d}X(t) - \left(\delta_1 1_{\{U_k(t) {>} b_1\}}+\delta_2 1_{\{U_k(t) {>} b_2\}}+...+\delta_k 1_{\{U_k(t) {>} b_k\}} \right)\mathrm{d}t , \quad t \geq 0 ,$$ where $X$ is a spectrally negative Lévy process, $\delta_1,...,\delta_k$ and $b_1{<}b_2{<}...{<}b_n$ are model parameters. Moreover, we present the formulas for one and two sided exit problems written in terms of the new $q$-scale functions associate with the process $U_k$. We also present new properties of the obtained scale functions. Finally, we extend the theory of multi-refracted processes to processes with general premium rate function $\phi$.

14:15-15:15 Noriyoshi Sakuma (Aichi University of Education)
Title: Unimodality for Lévy process $(X_t)_{t ¥geq 0}$ on $Z_+$: large time behavior
Abstract: In this talk, we consider Lévy process $(X_t)_{t¥geq 0}$ on $Z_+$. First, we shall give a brief survey of discrete infinitely divisible distributions. Second, we explain that discrete unimodality for Lévy process $(X_t)_{t¥geq 0}$ on $Z_+$. It was studied by T. Wanatabe in 90's and applied to Lévy process on $R_+$. Especially, he proved that for Lévy process $\displaystyle (X_t)_{t ¥ge 0}$ with a bounded discrete Lévy measure the law of $X_t$ becomes unimodal at large time. Recently, we show behavior of mode at large time. This talk is based on a joint work with Togawa.

15:30-16:30 Juan Carlos Pardo (CIMAT)
Title: Abrupt convergenece for generalized Ornstein-Uhlenbeck process
Abstract: In this talk, we study the cut-off phenomenon for a family of d-dimensional Ornstein-Uhlenbeck processes driven by Lévy processes. Under some suitable conditions on the drift matrix and the Lévy measure of the driven processes present a profile cut-off with respect to the total variation distance. This is a joint work with Gerardo Barrera.