We propose an analytically tractable class of models for the dynamics of a limit order book, described as the solution of a stochastic partial differential equation (SPDE) with multiplicative noise. We provide conditions under which the model admits a finite dimensional realization driven by a (low-dimensional) Markov process, leading to efficient methods for estimation and computation. We study two examples of parsimonious models in this class: a two-factor model and a model in which the order book depth is mean-reverting. For each model we perform a detailed analysis of the role of different parameters, study the dynamics of the price, order book depth, volume and order imbalance, provide an intuitive financial interpretation of the variables involved and show how the model reproduces statistical properties of price changes, market depth and order flow in limit order markets.

We establish numerical methods for solving the martingale optimal transport problem (MOT) - a version of the classical optimal transport with an additional martingale constraint on transport's dynamics. We prove that the MOT value can be approximated using linear programming (LP) problems which result from a discretisation of the marginal distributions combined with a suitable relaxation of the martingale constraint. Specialising to dimension one, we provide bounds on the convergence rate of the above scheme. We also show a stability result under only partial specification of the marginal distributions. Finally, we specialise to a particular discretisation scheme which preserves the convex ordering and does not require the martingale relaxation. We introduce an entropic regularisation for the corresponding LP problem and detail the corresponding iterative Bregman projection. We also rewrite its dual problem as a minimisation problem without constraint and solve it by computing the concave envelope of scattered data.

We describe the pricing and hedging practices refraining from the use of probability. We encode volatility in an enhancement of the price trajectory and we give pathwise presentations of the fundamental equations of Mathematical Finance. In particular this allows us to assess model misspecification, generalising the so-called fundamental theorem of derivative trading (see Ellersgaard et al. 2017). Our pathwise integrals and equations exhibit the role of Greeks beyond the leading-order Delta, and makes explicit the role of Gamma sensitivities.

We show how to set up a forward rate model in the presence of volatility uncertainty by using the theory of G-Brownian motion. In order to formulate the model, we extend the G-framework to integration with respect to two integrators and prove a version of Fubini's theorem for stochastic integrals. The evolution of the forward rate in the model is described by a diffusion process, which is driven by a G-Brownian motion. Within this framework, we derive a sufficient condition for the absence of arbitrage, known as the drift condition. In contrast to the traditional model, the drift condition consists of two equations and two market prices of risk, respectively, uncertainty. Furthermore, we examine the connection to short rate models and discuss some examples.

A key driver of household food insecurity is the balance between household disposable income and food prices. In the UK in recent years, many households have relied on charitable food banks in order to eat. The UK exit of the EU (Brexit) is expected to have significant influence across the economy, including on employment, wages and prices. We use structured expert judgement to quantify food prices and associated uncertainties in a principled manner under two scenarios: a Brexit deal broadly similar to the present situation and Brexit without a deal. Here we show that consumer prices index (CPI) food prices will rise faster than previously under both scenarios (6% and 24% respectively) and that the uncertainty is significantly larger under no-deal (a plausible worst-case of 43% compared to 14% ). Under a Brexit deal, a family with two adults and two children, one at pre-school and one at primary school, buying healthy diets will suffer median increases in food costs of {\pounds}6 per week (with a plausible worst case of {\pounds}16 per week). Under a no-deal scenario, this family will suffer a median increase of {\pounds}23 per week (plausible worst-case {\pounds}45 per week). The Lancet reported that poor diet is linked to 20% of all deaths worldwide, though association with disease. Rising food prices in a condition of static UK incomes is likely to add to demand pressure on the NHS. Our results demonstrate that uncertainty analysis associated with expected changes in food costs are vital to inform policies that allow households to afford a food basket that meets their needs.