In many situations where stochastic modeling is used, one desires to choose the coefficients of a stochastic differential equation which represents the reality as simply as possible. For example one desires to approximate a diffusion model
with high complexity coefficients by a model within a class of simple diffusion models. To achieve this goal, we introduce a new Wasserstein type distance on the set of laws of solutions to d-dimensional stochastic differential equations.
This new distance $\widetilde{W}^{2}$ is defined similarly to the classical Wasserstein distance $\widetilde{W}^{2}$ but the set of couplings is restricted to the set of laws of solutions of 2$d$-dimensional stochastic differential equations. We prove that this new distance $\widetilde{W}^{2}$ metrizes the weak topology. Furthermore this distance $\widetilde{W}^{2}$ is characterized in terms of a stochastic control problem. In the case d = 1 we can construct an explicit solution. The multi-dimensional case, is more tricky and classical results do not apply to solve the HJB equation because of the degeneracy of the differential operator. Nevertheless, we prove that this HJB equation admits a regular solution.[-]

We represent Hawkes process and their Volterra long term limits, which have recently been used as rough variance processes, as functionals of infinite dimensional affine Markov processes. The representations lead to several new views on affine Volterra processes considered by Abi-Jaber, Larsson and Pulido. We also discuss possible extensions to rough covariance modeling via Volterra Wishart processes.
The talk is based on joint work with Josef Teichmann.[-]

I discuss some recent developments related to the robust framework for pricing and hedging in discrete time. I introduce pointwise approach based on pathspace restrictions and compare it with the quasi-sure setting of Bouchard and Nutz (2015), and show that their versions of the Fundamental Theorem of Asset Pricing and the Pricing-Hedging duality may be deduced one from the other via a construction of a suitable set of paths which represents a given set of measures. I show that the setup with statically traded hedging instruments can be naturally lifted to a setup with only dynamically traded assets without changing the superhedging prices. This allows one to deduce, in particular, a pricing-hedging duality for American options. Subsequently, I focus on the superhedging problem and discuss the choice of a trading strategy amongst all feasible super-hedging strategies. First, I establish existence of a minimal superhedging strategy and characterise its value via a concave envelope construction. Then I introduce a secondary problem of maximisation of expected utility of consumption. Building on Nutz (2014) and Blanchard and Carassus (2017) I provide suitable assumptions under which an optimal strategy exists and is unique. Finally, I also explain how additional information can be seen as a further restriction of the pathspace. This allows one to quantify to value of such a new information. The talk is based on a number of recent works (see references) as well as ongoing research with Johannes Wiesel.[-]

(Joint work with Gonçalo Jacinto and Patricia A. Filipe.) The effect of random fluctuations of internal and external environmental conditions on the growth dynamics of individual animals is not captured by the regression model typical approach. We use stochastic differential equation (SDE) versions of a general class of models that includes the classical growth curves as particular cases. Namely, we use models of the form $d Y_t=\beta\left(\alpha-Y_t\right) d t+\sigma d W_t$, with $X_t$ being the animal size at age $t$ and $Y_t=h\left(X_t\right)$ being the transformed size by a $C^1$ monotonous function $h$ specific of the appropriate underlying growth curve model. $\alpha$ is the average transfomed maturity size of the animal, $\beta>0$ is the rate of approach to it and $\sigma>0$ measures the intensity of the effect on the growth rate of $Y_t$ of environmental fluctuations. These models can be applied to the growth of wildlife animals and also to plant growth, particularly tree growth, but, due to data availability (data furnished by the Associação dos Produtores de Bovinos Mertolengos - ACBM) and economica interest, we have applied them to cattle growth.
We briefly mention the extensive work of this team on parameter simulation methods based on data from several animals, including alternatives to maximum likelihood to correct biases and improve confidence intervals when, as usually happens, there is shortage of data for animals at older ages. We also mention mixed SDE models, in which model parameters may vary randomly from animal to animal (due, for instance, to their different genetical values and other individual characteristics), including a new approximate parameter estimation method. The dependence on genetic values opens the possibility of evolutionary studies on the parameters.
In our application to mertolengo cattle growth, the issue of profit optimization in cattle raising is very important. For that, we have obtained expressions for the expected value and the standard deviation of the profit on raising an animal as a function of the selling age for quite complex and market realistic raising cost structures and selling prices. These results were used to determine the selling age that maximizes the expected profit. A user friendly and flexible computer app for the use of farmers was developed by Ruralbit based on our results.[-]