An American option is a contract whereby one investor (the buyer) may receive a payment from another investor (the seller) at a time of his/her choosing (i.e. a stopping time). When pricing options in a classical complete financial model (without friction), one can usually obtain a dual representation of the price of an American option which involves a probability measure with respect to which the discounted stock price is a martingale, as well as a stopping time. When transaction costs enters the picture, it becomes necessary to look into larger classes of objects in order to obtain comparable dual representations, including so-called approximate martingales and mixed (or randomised) stopping times. I will also show what happens when we allow the buyer to use a mixed rather than an ordinary stopping time to exercise the option. The details will be presented in a simple (binary tree) model, and no prior knowledge of finance is required.

This talk is based on joint work with Tomasz Zastawniak and Chih-Yuan Tien.

What is the probability that the convex hull of a random walk in $\mathbb{R}^d$ does not absorb the origin by the time $n$? In dimension one this means that there is no sign change in the trajectory. The remarkable formula of Sparre Andersen (1949) states that any random walk with symmetrically and continuously distributed increments stays positive with probability $(2n-1)!!/(2n)!!$ regardless of the distribution. We prove a multidimensional analogue of this result providing an explicit tractable distribution-free formula for the absorption probability. The main idea is to show that the absorption problem is equivalent to the geometric problem on counting the number of Weyl chambers in $\mathbb{R}^n$ intersected by a generic linear subspace of codimension $d$. This method also applies to the absorption problem for convex hulls of random walk bridges and even for joint convex hulls of several symmetric random walks and/or bridges. Our distribution-free formulas result in a number of applications, including a new multidimensional form of the arsine law.

This talk is based on the joint works with Zakhar Kabluchko (Munster) and Dmitry Zaporozhets (St. Petersburg).

The connections between electricity and probability are deep, and have provided many tools for understanding the behaviour of stochastic processes. In this talk, I will describe a recent result in this direction, which states that if a sequence of spaces equipped with so-called "resistance metrics" and measures converge with respect to the Gromov-Hausdorff-vague topology, and a certain non-explosion condition is satisfied, then the associated stochastic processes also converge. This result generalises previous work on trees, fractals, and various models of random graphs. I further conjecture that it will be applicable to the random walk on the incipient infinite cluster of critical bond percolation on the high-dimensional integer lattice.

I shall begin by briefly motivating and introducing epidemic models where the population has some sort of random network structure. Then I will introduce a couple of particular random network models which are sufficiently complex to incorporate some desirable network features, yet simple enough that a stochastic SIR (susceptible $\to$ infective $\to$ removed/recovered) epidemic upon the network submits to asymptotic analysis of many of its final size properties. I will also address the issue of incorporating vaccination into these models.