## Probability in the North East workshop

#### 21 January 2021

Organizers: Matthew Aldridge and Martín López-García (Leeds).

The virutal meeting will run for half a day (11.30–17.00, precise schedule will be shared closer to the date). We are very happy to have the following speakers for this meeting:

• Sebastian Herrmann (University of Manchester)
• Robin Stephenson (University of Sheffield)
• Polly-Anne Jeffrey (University of Leeds)
• Roberta Merli (University of York)
• Chak Hei (Hugo) Lo (University of Edinburgh)
• Conrado da Costa (Durham University)

The meeting will be virtual and registration is free, but please do register by sending an email to Martin Lopez-Garcia (m.lopezgarcia@leeds.ac.uk) by Friday 15th January. The details for joining the meeting will be sent to those registered closer to the date.

## Programme

We consider a particle moving linearly inside a curved conic region ($R = \{(x,y) : 0 \leq y \leq x^\gamma \}$, $|\gamma| <1$) with Markovian reflection at the boundary. The purpose of this talk is to determine, with respect to the reflection law, the values of $\gamma$ for which the process is recurrent or transient.

This is a joint work in progress with Mikhail Menshikov and Andrew Wade.
Polly-Anne Jeffrey (University of Leeds)
Cellular receptors which reside in the cell membrane can bind with ligand molecules in the extracellular medium to form monomers. These interactions ultimately determine the fate of a cell through intracellular processes which differ depending on the receptor and ligand combination. Often, different receptors can bind a common ligand, where the outcome of this interaction differs according to the receptor type. Hence, there is an element of natural competition for a common ligand between receptors of different types and the dynamics can be modelled mathematically as a competition process. Here I will introduce a stochastic competition process which is appropriate for modelling scenarios in which there are low copy numbers of molecules. I will present two methods of analysis of stochastic descriptors for such a process, one a matrix analytic approach and the other a “moderate competition" approximation.
12:45–13:30
Lunch
Roberta Merli (University of York)
Consider the process that diffuses like a Brownian motion on the circumference of the unit circle and, at times given by an independent Poisson process, jumps to the opposite point on the circle.

We consider ways in which two copies of this process may be coupled, and use excursion theory and Bellman's optimality principle to determine the coupling which minimises the expected coupling time amongst all co-adapted couplings. The optimal coupling strategy turns out to depend intuitively upon the rate of the driving Poisson process.
14:00–14:45
Sebastian Herrmann (University of Manchester)
We consider the martingale optimal transport duality for càdlàg processes with given initial and terminal laws. Strong duality and existence of dual optimizers (robust semi-static superhedging strategies) are proved for a class of payoffs that includes American, Asian, Bermudan, and European options with intermediate maturity. We exhibit an optimal superhedging strategy for which the static part solves an auxiliary problem and the dynamic part is given explicitly in terms of the static part.

This talk is based on joint work with Florian Stebegg.
14:45–15:00
Break
Chak Hei Lo (University of Edinburgh)
In this presentation we will look at the cutpoints, a separation property of trajectories, of transient discrete-time stochastic processes in ${\mathbb R}_+$ and ${\mathbb R}^d$. A point $x$ of ${\mathbb R}_+$ is a cutpoint for a given trajectory of a stochastic process if, roughly speaking, the process visits $x$ and never returns to $[0,x)$ after its first entry into $(x,\infty)$. A similar notion is applicable in higher dimensions. Under mild conditions, cutpoints may appear only in the transient case, when trajectories escape to infinity. A fundamental question is: does a transient process have infinitely many cutpoints, or not? We give conditions under which near-critical stochastic processes on the half-line have infinitely many or finitely many cutpoints, generalizing existing results on nearest-neighbour random walks to processes with bounded increments satisfying appropriate increment moments conditions. We apply one of these results to deduce that a class of transient zero-drift Markov chains in ${\mathbb R}^d$, $d \geq 2$, possess infinitely many separating annuli, generalizing previous results on spatially homogeneous random walks.

This is joint work with Mikhail V. Menshikov and Andrew R. Wade.
Robin Stephenson (University of Sheffield)
We consider the random directed graph $\vec{G}(n,p)$ with vertex set $\{1,2,\ldots,n\}$ in which each of the $n(n-1)$ possible directed edges is present independently with probability $p$. We are interested in the strongly connected components of this directed graph. A phase transition for the emergence of a giant strongly connected component is known to occur at $p = 1/n$, with critical window $p= 1/n + \lambda n^{-4/3}$ for $\lambda \in \mathbb{R}$. We show that, within this critical window, the strongly connected components of $\vec{G}(n,p)$, rescaled by $n^{-1/3}$, converge in distribution to a sequence $(\mathcal{C}_1,\mathcal{C}_2,\ldots)$ of finite strongly connected directed multigraphs with edge lengths which are either 3-regular or loops. The convergence occurs the sense of an $\ell^1$ sequence metric for which two directed multigraphs are close if there are compatible isomorphisms between their vertex and edge sets which roughly preserve the edge-lengths. Our proofs rely on a depth-first exploration of the graph which enables us to relate the strongly connected components to a particular spanning forest of the undirected Erdős–Rényi random graph $G(n,p)$, whose scaling limit is well understood. We show that the limiting sequence $(\mathcal{C}_1,\mathcal{C}_2,\ldots)$ contains only finitely many components which are not loops.

Joint work with Christina Goldschmidt.