### Department of Mathematics and Systems Analysis

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The talks are held in lecture hall U3 at Aalto University, Otakaari 1.

Tentative schedule:

10.00-10.45 | Alvarez |

11.00-11.45 | Manolescu |

12.00-13.00 | [lunch] |

13.00-13.45 | Myllymäki |

14.00-14.45 | Geiss |

14.45-15.15 | [coffee] |

15.15-16.00 | Proutiere |

16.15-17.00 | Webb |

17:30- | [sauna] |

We consider the determination of the optimal stationary singular stochastic control of a linear diffusion for a class of average cumulative cost minimization problems arising in various financial and economic applications of stochastic control theory. We present a set of conditions under which the optimal policy is of the standard local time reflection type and state the first order conditions from which the boundaries can be determined. Since the conditions do not require symmetry or convexity of the costs, our results cover also the cases where costs are asymmetric and non-convex. We also investigate the comparative static properties of the optimal policy and delineate circumstances under which higher volatility expands the continuation region where utilizing the control is suboptimal.

I will discuss global envelope tests which are a graphical and statistically rigorous tool for comparing an empirical function with its simulated counterparts under a null model. We originally developed global envelope tests for goodness-of-fit testing in spatial statistics, but generally speaking the proposed global envelope tests are tests of a hypothesis on the functions or multivariate vectors and they can be applied to any functional or multivariate data. Thus, they have applications, e.g., in functional data analysis as well. I will present how the global envelopes can be used, for example, for testing hypothesis in spatial statistics and for testing the equality of means of groups of functions (functional ANOVA). The advantage of the global envelope tests are that they provide a graphical interpretation of the test results.

We consider an approximation problem for stochastic integrals which occurs in Stochastic Finance while discrete time hedging of European options. Analysing the local approximation error of this approximation yields to Riemann-Liouville operators, fractional gradient processes,and spaces of weighted bounded mean oscillation. This work develops further [S. Geiss, Prob. Theory Related Fields 132, pp.39-73, 2005] and [C. Geiss, S. Geiss, and E. Laukkarinen, Potential Analysis 39, pp.203-230, 2013]. Joint work with Thuan Nguyen (University of Jyväskylä).

In this talk, we consider cluster detection in Block Markov Chains. These Markov chains are characterized by a block structure in their transition matrix. More precisely, the n possible states are divided into a finite number of K groups or clusters, such that states in the same cluster exhibit the same transition rates to other states. One observes a trajectory of the Markov chain, and the objective is to recover, from this observation only, the (initially unknown) clusters. We devise a clustering procedure that accurately, efficiently, and provably detects the clusters. We first derive a fundamental information-theoretical lower bound on the detection error rate satisfied under any clustering algorithm. This bound identifies the parameters of the Block Markov Chain and trajectory lengths, for which it is possible to accurately detect the clusters. We next develop two clustering algorithms that can together accurately recover the cluster structure from the shortest possible trajectories, whenever the parameters allow detection. These algorithms thus reach the fundamental detectability limit, and are optimal in that sense. Joint work with Jaron Sanders (U. Delft) and S. Yun (KAIST)

I will discuss a general approach to multivariate normal approximation, which is inspired by Stein's method. As an application of this, I will discuss a central limit theorem in the setting of of random matrix theory. This is based on joint work with Gaultier Lambert from Zürich and Michel Ledoux from Toulouse.

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