Week 27.10.2025 – 02.11.2025

The Rademacher random walk associated with a deterministic sequence $(a_n)_{n \geq 1}$ is the walk which starts at zero and, at step $i$, independently steps either up or down by $a_i$ with equal probability. We will consider the following natural questions: what conditions on the growth rate of $(a_n)$ guarantee that the walk is transient? What growth conditions guarantee that the walk is weakly recurrent?

We will show that if the sequence is bounded, the walk is weakly recurrent, while if $a_n = n^{\alpha + o(1)}$ for some $\alpha > 1/2$, the walk is transient. We will also see that both of these results are in some sense tight.

The talk is based on joint work with Satyaki Bhattacharya and Ed Crane.

In this talk, we present a pathwise approach to stochastic
integration: Using the concepts of quadratic variation and Lévy area of
a continuous path along a sequence of time partitions, we construct a
pathwise integral as a limit of general Riemann sums. In a probabilistic
framework, when the underlying process is a semimartingale, this notion
of integration is consistent with stochastic integration. Furthermore,
we state necessary and sufficient conditions for the quadratic variation
and Lévy area of a continuous path to be invariant with respect to the
choice of the partition sequence.

We show that the Markov fractions introduced recently by Boris Springborn are precisely the slopes of the exceptional vector bundles on $\mathbb P^2$ studied in 1980s by Dr\`ezet and Le Potier and by Rudakov. In particular, we provide a simpler proof of Rudakov's result claiming that the ranks of the exceptional bundles on $\mathbb P^2$ are Markov numbers.

We analyze the stability of financial investment networks, where financial institutions hold overlapping portfolios of assets. We consider the effect of portfolio diversification and heterogeneous investments using a random matrix dynamical model driven by portfolio rebalancing. The stability/instability transition is dictated by the largest eigenvalue of the random matrix governing the time evolution of the endogenous components of the returns, for which we will propose two different approximation schemes, and finally explore the replica method calculations that underpin one of the approximation schemes

I will discuss two pieces of work. In the first we present theoretical results related to preconditioning Markov chain Monte Carlo sampling algorithms. Preconditioning is a widely used technique that is known empirically to improve the mixing of Markov chain algorithms, but little has been said theoretically about it. I will present some recent work establishing some positive and negative theoretical results. I will then discuss some methodological work devising new preconditioners. The standard options to choose from in common software packages are 'diagonal' or 'dense'. We will present a new alternative option that seeks to improve upon diagonal preconditioning whilst also being less computationally expensive than the quadratic cost required for the dense option. Both projects are joint work with my former PhD student Max Hird, now a PDRA at University of Waterloo. The first is associated with this paper: https://www.jmlr.org/papers/v26/23-1633.html.

Among the different methods proposed to lift path-dependent dynamics to infinite-dimensional Markovian frameworks, the use of signatures appears especially natural, as linear functionals of the signature can approximate any continuous path functional (with respect to suitable Hölder/variation topologies) arbitrarily well. While such universal approximation results at the level of the vector fields are well established, we go further and consider solutions of generic path-dependent controlled differential equations (CDEs). We then show that, under mild regularity assumptions, any such path-dependent system can be approximated by a suitable signature CDE. To this end we first establish well-posedness and stability of path-dependent systems using weighted space topologies for Hölder continuous paths. We then transfer these results to signature CDEs, deriving in particular well-posedness conditions and a dynamic universal approximation theorem when the vector fields are real-analytic functions of the signature.

This talk is based on joint work with Tomas Carrondo, Paul Hager, and Fabian Harang.

TBD