Department of Mathematics and Systems Analysis

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

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]

Luis Alvarez Esteban (University of Turku)
A Class of Solvable Stationary Singular Stochastic Control Problems of Linear Diffusions

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.
Ioan Manolescu (Université de Fribourg)
Uniform Lipschitz functions on the triangular lattice have logarithmic variations
Uniform integer-valued Lipschitz functions on a finite domain of the triangular lattice are shown to have variations of logarithmic order in the radius of the domain. The level lines of such functions form a loop O(2) model on the edges of the hexagonal lattice with edge-weight one. An infinite-volume Gibbs measure for the loop O(2) model is constructed as a thermodynamic limit and is shown to be unique. It contains only finite loops and has properties indicative of scale-invariance: macroscopic loops appearing at every scale. The existence of the infinite-volume measure carries over to height functions pinned at 0; the uniqueness of the Gibbs measure does not. The proof is based on a representation of the loop O(2) model via a pair of spin configurations that are shown to satisfy the FKG inequality. We prove RSW-type estimates for a certain connectivity notion in the aforementioned spin model. Based on joint work with Alexander Glazman. 

Mari Myllymäki (Natural Resources Institute Finland (Luke))
Global envelopes for testing with functional test statistics and functional data analysis

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.

Stefan Geiss (University of Jyväskylä)
Approximation of stochastic integrals, Riemann-Liouville operators, and bounded mean oscillation

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).

Alexandre Proutiere (KTH, Stockholm)
Clustering in Block Markov Chains

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)

Christian Webb (Aalto University)
When is a random variable close to being normally distributed?

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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