Stochastic Sauna '22
- Jonas Tölle, Aalto University, Finland.
- Lauri Viitasaari, Uppsala University, Sweden.
- Tapio Helin, LUT University, Finland.
- Teemu Pennanen, King's College London, UK.
- Trishen Gunaratnam, University of Geneva, Switzerland.
- Orlando Margiliano, KTH Royal Institute of Technology, Sweden.
University of Helsinki, Porthania P674, Yliopistonkatu 3, Helsinki.
The lecture hall is equipped with a blackboard, beamer, and a computer.
The social program includes a sauna and dinner. Sauna in Löyly at 18.00 after the seminar, the entry is offered by the seminar organization (bring your swimsuit) and you will have to pay the dinner by yourself.
There is no participation fee, but registration is mandatory. Please fill in the registration form.
The presentations are 40 minutes of talk plus 5 min of questions.
9:30-10:15 Singular limits for stochastic equations by J. Tölle.
Abstract: We study singular limits of nonlinear stochastic evolution equations in the interplay of disappearing strength of the noise and increasing roughness of the noise, so that the noise in the limit would be too rough to define a solution to the limiting equations. Simultaneously, the limit is singular in the sense that the leading order differential operator may vanish. Although the noise is disappearing in the limit, additional deterministic terms appear due to renormalization effects. We give an abstract framework for the main error estimates, that first reduce to bounds on a residual and in a second step to bounds on the stochastic convolution. Moreover, we apply it to a singularly regularized Allen-Cahn equation and the Cahn-Hilliard/Allen-Cahn homotopy. See https://arxiv.org/abs/2204.09545
Joint work with Dirk Blömker, University of Augsburg, Germany.
10:20-10:50 Coffee break
10:55-11:40 Classifying one-dimensional discrete statistical models with maximum likelihood degree one by O. Margiliano.
Abstract: A statistical model is a set of probability distributions. When this set also has the structure of an algebraic variety, it becomes an object of study in algebraic statistics. In this talk I illustrate the interplay of statistics, algebraic geometry, and combinatorics by presenting a classification problem for the class of models named in the title. I describe a partial solution of this problem using algebraic techniques and the newly introduced combinatorial game of 'splitting chips on a grid'.
This talk is based on joint work with Arthur Bik.
11:45-12:30 Statistical inverse learning and regularization by projections by T. Helin.
Abstract: Statistical inverse learning aims at recovering an unknown function f from randomly scattered and possibly noisy point evaluations of another function g, connected to f via an ill-posed mathematical model. In this talk I blend statistical inverse learning theory with the classical regularization strategy of applying finite-dimensional projections and derive estimators that are provably minimax optimal.
12:30-14:00 Lunch break
14:00-14:45 On the existence and regularity of local times by L. Viitasaari.
Abstract: In this talk we study the existence and regularity of local times for general d-dimensional stochastic processes. We begin with a brief introduction to local times and discuss several prominent applications they have. As the main results, we give a general condition for the existence of local times with certain regularity properties. Various examples are provided including Gaussian quasi-helices, the Rosenblatt process, and solutions to SDEs driven by the fractional Brownian motion.
This talk is based on a joint work with T. Sottinen and E. Sönmez.
14:50-15:35 Tangled random currents and the continuity of phase transition for $\phi^4$ by T. Gunaratnam.
Abstract: The $\phi^4$ model is a classical model of ferromagnetism in statistical mechanics that has its origins in Euclidean quantum field theory. It is amongst the simplest models of unbounded spin that is expected to be in the Ising universality class. In this talk, I will describe recent progress made with my collaborators Christoforos Panagiotis (Université de Genève), Romain Panis (Université de Genève), and Franco Severo (ETH Zürich) towards understanding the critical behaviour of $\phi^4$ on the hypercubic lattice $\mathbb{Z}^d$ for $d\geq 3$. In particular, we will discuss a new geometric representation for $\phi^4$ called random tangled currents. This is the natural extension of the celebrated random current representation for the Ising model that has been the basis of many beautiful results concerning its critical behaviour since the seminal work of Aizenman in the ‘80s. At the heart of these results is the switching lemma. We prove that random tangled currents satisfy an analogous switching principle. If time permits I will discuss the proof of continuity of the phase transition for $\phi^4$ in $d\geq 3$, focusing on how to adapt percolation-based techniques to our setting, where there are significant conceptual and technical difficulties arising from the unboundedness of spin and lack of exact combinatorial symmetries in the model.
15:40-15:45 Break
16:00-16:45 Convex stochastic optimization by T. Pennanen.
Abstract: We study dynamic programming, duality and optimality conditions in general convex stochastic optimization problems introduced by Rockafellar and Wets in the 70s. We give a general formulation of the dynamic programming recursion and derive an explicit dual problem in terms of two dual variables, one of which is the shadow price of information while the other one gives the marginal cost of a perturbation much like in classical Lagrangian duality. Existence of primal solutions and the absence of duality gap are obtained without compactness or boundedness assumptions. In the context of financial mathematics, the relaxed assumptions are satisfied under the well-known no-arbitrage condition and the reasonable asymptotic elasticity condition of the utility function. We extend classical portfolio optimization duality theory to problems of optimal semi-static hedging. Besides financial mathematics, we obtain several new results in stochastic programming and stochastic optimal control.
The workshop is hosted by the Department of Mathematical and Statistical Sciences, University of Helsinki, and organized by
Dario Gasbarra, dario.gasbarra@helsinki.fi
Gerardo Barrera, gerardo.barreravargas@helsinki.fi
Nikolay Barashkov, nikolay.barashkov@helsinki.fi