Stochastic analysis and stochastic processes

Organizadores: Paavo Salminen (paavo.salminen@abo.fi), Antoine Lejay (Antoine.Lejay@univ-lorraine.fr), Ernesto Mordecki (mordecki@cmat.edu.uy)

  • Miércoles 15

    15:00 - 15:45

    Diffusion through a two-sided membrane

    Ilya Pavlyukevich (Universität Jena, Alemania)

    We study the Markov chains on \(\mathbb{Z}^d\), \(d\geq 2\), that behave like a standard symmetric random walk outside of the hyperplane (membrane) \(H=\{0\}\times \mathbb{Z}^{d-1}\). The transition probabilities on the membrane \(H\) are periodic and also depend on the incoming direction to \(H\), that makes the membrane \(H\) two-sided. Moreover, sliding along the membrane is allowed. We show that the natural scaling limit of such Markov chains is a \(d\)-dimensional diffusion whose first coordinate is a skew Brownian motion and the other \(d-1\) coordinates is a Brownian motion with a singular drift controlled by the local time of the first coordinate at \(0\). In the proof we utilize a novel martingale characterization of the Walsh Brownian motion and determine the effective permeability and slide direction. Eventually, a similar convergence theorem is established for the one-sided membrane without slides and random iid transition probabilities.

    Based on joint work with A. Pilipenko (Institute of Mathematics, NAS of Ukraine, Kiev).

    15:45 - 16:30

    Diffusion spiders: resolvents, excessive functions and optimal stopping

    Paavo Salminen (Åbo Akademi University, Finlandia)

    Let \(\Gamma\) be a graph with one internal node \(\mathbf{0}\) and \(n\geq 1\) edges meeting each other at \(\mathbf{0}\). For \(x>0\) and \(i\in\{1,2,\dots,n\}\) we let \(\mathbf{x}=(x,i)\) denote the point on \(\Gamma\) located on the \(i\)'th edge at the distance \(x\) from \(\mathbf{0}\).

    Let \(X=(X_t)_{t\geq 0}\) be a one-dimensional diffusion on \(\mathbf{R}_+\) hitting 0 with probability one. A (homogeneous) diffusion spider \(\mathbf{X}=(\mathbf{X}_t)_{t\geq 0}\) is a continuous strong Markov process on \(\Gamma\) which on each edge before hitting \(\mathbf{0}\) behaves as \(X\) before hitting 0. When reaching \(\mathbf{0}\) the spider chooses, roughly speaking, an edge with some given probability to continue the movement. A rigorous construction of \(\mathbf{X}\) can be done using the excursion theory.

    In this talk we give an explicit expression for the resolvent kernel of \(\mathbf{X}\). Using this we study the excessive functions of \(\mathbf{X}\) and derive the so called glueing condition to be satisfied at \(\mathbf{0}\). We apply the representation theory (Martin boundary theory) of excessive functions and calculate the representing measure of a given excessive function.

    This machinery can be used to solve optimal stopping problems for \(\mathbf{X}\). As an example we consider Walsh's Brownian spider where the diffusion \(X\) is a Brownian motion. We focus on the case where the reward is continuous at \(\mathbf{0}\), linear on the edges and such that the resulting stopping region is connected.

    Based on joint work with Jukka Lempa and Ernesto Mordecki.

    16:45 - 17:30

    Non-zero-sum optimal stopping game with continuous versus periodic observations

    José Luis Pérez Garmendia (Centro de Investigación en Matemáticas, México)

    We introduce a new non-zero-sum game of optimal stopping with asymmetric information. Given a stochastic process modelling the value of an asset, one player has full access to the information and observes the process completely, while the other player can access it only periodically at independent Poisson arrival times. The first one to stop receives a reward, different for each player, while the other one gets nothing. We study how each player balances the maximisation of gains against the maximisation of the likelihood of stopping before the opponent. In such a setup, driven by a Lévy process with positive jumps, we not only prove the existence, but also explicitly construct a Nash equilibrium with values of the game written in terms of the scale function.

    Based on joint work with Kazutoshi Yamazaki and Neofytos Rodosthenous.

    17:30 - 18:15

    Optimal couplings for interacting particle systems via optimal transport

    Joaquin Fontbona (Universidad de Chile, Chile)

    We will review some ideas recently developed in order to obtain optimal convergence rates for mean-field stochastic particle systems interacting through binary jumps, by constructing trajectorial couplings that make use of optimal transport plans for empirical measures. Examples of applications in kinetic theory and in branching population models will be discussed.

    Based on joint works with Roberto Cortez and Felipe Muñoz.

  • Jueves 16

    15:00 - 15:45

    Sufficient variability of paths and differential equations with BV-coefficients

    Lauri Viitasaari (Aalto University, Finlandia)

    Partial differential equations and differential systems play a fundamental role in many aspects of our daily life. However, in many applications, especially in the field of stochastic differential or partial differential equations, the underlying equation does not make sense in a classical way, and one has to consider integral equations instead. Moreover, even the concept of integral is subtle. For example, a typical situation is that one needs to consider \(\int_0^t \varphi(X_s)\,dY_s\), where \(\varphi\) is a given function and \(X,Y\) are some continuous but non-differentiable objects. Several powerful techniques such as rough path theory have emerged to treat these situations, and it can be safely stated that nowadays differential systems driven by rough signals are already rather well understood. The idea in the rough path theory is, roughly speaking, that one assumes additional smoothness on the function \(\varphi\) in order to compensate bad behaviour of signal functions \(X\) and \(Y\). As such, the methodology cannot be applied in any straightforward manner if one allows discontinuities in \(\varphi\). In particular, this is the case if \(\varphi\) is a general BV-function.

    In this talk we combine tools from fractional calculus and harmonic analysis to some fine properties of BV-functions and maximal functions, allowing us to give a meaningful definition for (multidimensional) integrals \(\int_0^t \varphi(X_s)\,dY_s\) with a BV-function \(\varphi\), provided that the functions \(X\) and \(Y\) are regular enough in the Hölder sense. Here enough regularity means better Hölder regularity than what is customary assumed in the rough path theory, and this gain in regularity can be used to compensate ill behaviour of \(\varphi\). The key idea is that the signal \(X\) should not spend too much time, in some sense, on the bad regions of \(\varphi\). We quantify this in terms of potential theory and Riesz energies. We also discuss several consequences, and provide existence and uniqueness results for certain differential systems involving BV-coefficients. Extensions and further topics are discussed.

    Based on joint work with Michael Hinz (Bielefeld University) and Jonas Tölle (University of Helsinki).

    15:45 - 16:30

    Estimating the parameter of the Skew Brownian motion

    Antoine Lejay (Institut Élie Cartan de Lorraine, Francia)

    The Skew Brownian motion may be constructed by choosing randomly the sign of a Brownian excursion. It thus depends on a parameter \(\theta\in[-1,1]\). In this talk, we discuss the statistical properties of the maximum likelihood estimator (MLE) for this parameter \(\theta\). In particular, it is consistent and asymptotically mixed normal as the local time of the process is involved in the limit, yet with a rate \(1/4\) and not \(1/2\) as usual. We also discuss some non-asymptotic properties of the MLE.

    Based on joint work with Sara Mazzonetto (IECL, Université de Lorraine, Nancy).

    16:45 - 17:30

    On the consistency of the least squares estimator in models sampled at random times driven by long memory noise

    Soledad Torres (Universidad de Valparaiso, Chile)

    In numerous applications data are observed at random times. Our main purpose is to study a model observed at random times incorporating a long memory noise process with a fractional Brownian Hurst exponent \(H\). In this talk, we propose a least squares (LS) estimator in a linear regression model with long memory noise and a random sampling time. The strong consistency of the estimator is established, with a convergence rate depending on \(N\) and Hurst exponent. A Monte Carlo analysis supports the relevance of the theory and produces additional insights.