%---------------------------------------------------------------------------% An afternoon in Probability: Torino meets Leeds Torino, April 16, 2019 %----------------------------------------------------------------------------%

Room 09 - ESOMAS Department Corso Unione Sovietica 218/bis, 10134, Torino

-----------------Program (Speakers and abstracts)------------------

14:30-15:10 Tiziano De Angelis (School of Mathematics, University of Leeds)

Title: Probabilistic results concerning smoothness of the value function and of the free boundary in optimal stopping

Abstract: I will present probabilistic proofs of some regularity properties for the value function of general optimal stopping problems and for the associated optimal boundaries. In particular this talk focuses on C^1 regularity of the value function and Lipschitz continuity of the optimal boundary. Most of our arguments rely on fundamental concepts from the theory of Markov processes and bridge the probabilistic and the analytical strands of the literature on free boundary problems. I will also illustrate situations in which our work improves or complements known facts from PDE theory.

15:10-15:40 Elena Issoglio (School of Mathematics, University of Leeds)

Title: A numerical scheme for a multidimensional SDEs with distributional drift

Abstract: This talk focuses on a multidimensional SDE where the drift is an element of a fractional Sobolev space with negative order, hence a distribution. This SDE admits a unique weak solution in a suitable sense - this was proven in [Flandoli, Issoglio, Russo (2017)]. The aim here is to construct a numerical scheme to approximate this solution. One of the key problems is that the drift cannot be evaluated pointwise, hence we approximate it with suitable functions using Haar wavelets, and then apply (an extended version of) Euler-Maruyama scheme. We then show that the algorithm converges in law, and in the special 1-dimensional case we also get a rate of convergence. This talk is based on a joint work with T. De Angelis and M. Germain.

15.40-16:10 coffee break

16:10-16:50 Elena Vigna (ESOMAS , University of Leeds)

Title: On time consistency for mean-variance portfolio selection

Abstract: This talk addresses a comparison between different approaches to time inconsistency for the mean-variance portfolio selection problem. We define a suitable intertemporal preferences-driven reward and use it to compare the three possible approaches to time inconsistency for the mean-variance portfolio selection problem over $[t_0,T]$: precommitment approach, consistent planning or game theoretical approach, and dynamically optimal approach. We find that the precommitment strategy beats the other strategies if the investor only cares at the view point at time $t_0$ and is not concerned to be time-inconsistent in $(t_0,T)$; the consistent planning strategy dominates the dynamically optimal strategy until a time point $t^*\in(t_0,T)$ and is dominated by the dynamically optimal strategy from $t^*$ onwards.

16:50-17:30 Cristina Zucca (Dept. of Mathematics, University of Torino)

Title: Inverse First Passage Time problem for diffusions

Abstract: In several applications the dynamics of the variables of interest is described via suitable stochastic processes constrained by boundaries and the focus is on the first passage time (FPT). This is the direct FPT problem. However, there are also instances when the underlying stochastic process is assigned, one knows or estimates the FPT distribution and wishes to determine the corresponding boundary shape. This is the inverse first passage time (IFPT) problem. Here we study this problem in the case of one dimensional diffusion process constrained by a single boundary or two boundaries. We also extend the IFPT method to multivariate Gauss-Markov processes and we investigate the boundary shape corresponding to given FPT distributions for suitable choices of the parameters. Special attention is given to the Integrated Brownian motion and the two-dimensional Ornstein-Uhlenbeck process. Some examples of applications of the proposed methods will be illustrated.

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