We consider the problem faced by a central bank which bails out distressed financial institutions that pose systemic risk to the banking sector. In a structural default model with mutual obligations, the central agent seeks to inject a minimum amount of cash to a subset of the entities in order to limit defaults to a given proportion of entities. We prove that the value of the agent's control problem converges as the number of defaultable agents goes to infinity, and it satisfies  a drift controlled version of the supercooled Stefan problem. We compute optimal strategies in feedback form by solving numerically a forward-backward coupled system of PDEs. Our simulations show that the agent's optimal strategy is to subsidise banks whose asset values lie in a non-trivial time-dependent region. Finally, we study a linear-quadratic version of the model where instead of the losses, the agent optimises a terminal loss function of the asset values. In this case, we are able to give semi-analytic strategies, which we again illustrate numerically. Joint work with Christa Cuchiero and Stefan Rigger.

We introduce a method for estimating the roughness of a function based on a discrete sample, using the concept of normalized p-th variation along a sequence of partitions. We discuss the consistency of this estimator in a pathwise setting under high-frequency asymptotics. We investigate its finite sample performance for measuring the roughness of sample paths of stochastic processes using detailed numerical experiments based on sample paths of Fractional Brownian motion and other fractional processes.
We then apply this method to estimate the roughness of realized volatility signals based on high-frequency observations.
Through a detailed numerical experiment based on a stochastic volatility model, we show that even when instantaneous volatility has diffusive dynamics with the same roughness as Brownian motion, the realized volatility exhibits rougher behaviour corresponding to a Hurst exponent significantly smaller than 0.5. Similar behaviour is observed in financial data, which suggests that the origin of the roughness observed in realized volatility time-series lies in the `microstructure noise' rather than the volatility process itself.

TBA

Many problems in engineering can be understood as controlling the bifurcation structure of a given device. For example, one may wish to delay the onset of instability, or bring forward a bifurcation to enable rapid switching between states. In this talk, we will describe a numerical technique for controlling the bifurcation diagram of a nonlinear partial differential equation by varying the shape of the domain. Our aim is to delay or advance a given branch point to a target parameter value. The algorithm consists of solving a shape optimization problem constrained by an augmented system of equations, called the Moore–Spence system, that characterize the location of the branch points. We will demonstrate the effectiveness of this technique on several numerical experiments on the Allen–Cahn, Navier–Stokes, and hyperelasticity equations.

The study of gravity currents has long been of interest due to their prevalence in industry and in nature, one such example being the spreading of viscoplastic (yield-stress) fluid films. When a viscoplastic fluid is extruded onto a flat plate, the resulting gravity current expands axisymmetrically when the surface is dry and rough. In this talk, I will discuss two instabilities that arise when (1) the no-slip surface is replaced by a free-slip surface; and (2) the flat plate is wet by a thin coating of water.

The injection of CO2 into porous subsurface reservoirs is a technological means for removing anthropogenic emissions, which relies on a series of complex porous flow properties. During injection of CO2 small-scale heterogeneities, often in the form of sedimentary layering, can play a significant role in focusing the flow of less viscous CO2 into high permeability pathways, with large-scale implications for the overall motion of the CO2 plume. In these settings, capillary forces between the CO2 and water preferentially rearrange CO2 into the most permeable layers (with larger pore space), and may accelerate plume migration by as much as 200%. Numerous factors affect overall plume acceleration, including the structure of the layering, the permeability contrast between layers, and the playoff between the capillary, gravitational and viscous forces that act upon the flow. However, despite the sensitivity of the flow to these heterogeneities, it is difficult to acquire detailed field measurements of the heterogeneities owing to the vast range of scales involved, presenting an outstanding challenge. As a first step towards tackling this uncertainty, we use a simple modelling approach, based on an upscaled thin-film equation, to create ensemble forecasts for many different types and arrangements of sedimentary layers. In this way, a suite of predictions can be made to elucidate the most likely scenarios for injection and the uncertainty associated with such predictions. 

I'll consider recent results concerning the stability of the classic Brunn-Minkowski inequality. In particular, I will focus on the linear stability for homothetic sets. Resolving a conjecture of Figalli and Jerison, we showed there are constants $C,d>0$ depending only on $n$ such that for every subset $A$ of $\mathbb{R}^n$ of positive measure, if $|(A+A)/2 - A| \leq d |A|$, then $|co(A) - A| \leq C |(A+A)/2 - A|$ where $co(A)$ is the convex hull of $A$. The talk is based on joint work with Hunter Spink and Marius Tiba.